The Riemann integral is the first rigorous construction of the [integral](/page/Integral) — the formalisation of the idea that the area under a curve can be captured as the [limit](/page/Limit) of a sum. It applies to bounded [functions](/page/Function) on closed intervals $[a, b] \subset \mathbb{R}$ and works by trapping the function between upper and lower approximations built from rectangles. When these approximations can be made arbitrarily close, the common value they converge to is the integral. The Riemann integral handles continuous functions, monotone functions, and more generally all bounded functions whose set of discontinuities is small in a precise measure-theoretic sense, but it breaks down for highly irregular functions and does not interact well with pointwise limits of [sequences](/page/Sequence) — limitations that eventually led to the Lebesgue theory. ## Motivation [motivation] ### The Naive Approach: Counting Squares The intuitive notion of "area under a curve" is ancient, but making it precise is harder than it appears. Given a non-negative function $f: [a, b] \to \mathbb{R}$, one might try to define the area of the region below the graph by overlaying a grid of small squares and counting how many fall inside. This approach has two fatal defects. First, grid squares that straddle the [boundary](/page/Boundary) of the region are ambiguous — they are partly inside and partly outside, and the finer the grid, the more boundary squares there are relative to area gained. Second, this method gives no algebraic handle on the area: there is no formula relating the grid count to the function $f$ itself, so computing the area of even simple regions (a triangle, a parabolic segment) requires a separate geometric argument each time. What is needed is a definition that connects the area directly to the values of $f$ and that comes with a built-in approximation scheme. ### The Antiderivative Approach A second natural idea is to bypass area altogether and define the integral through antiderivatives: declare $\int_a^b f(x)\, dx := F(b) - F(a)$ where $F' = f$. This is computationally powerful — it reduces integration to the problem of finding a primitive — but it has serious conceptual and technical deficiencies. Not every bounded function has an antiderivative; the function $\mathbb{1}_{[0,1]}$ has no differentiable primitive whose derivative equals $\mathbb{1}_{[0,1]}$ at the endpoints. Even when antiderivatives exist, they are not unique (any two differ by a constant) and there is no guarantee that an antiderivative can be found in closed form. More fundamentally, taking $F(b) - F(a)$ as the *definition* of the integral makes the [Fundamental Theorem of Calculus](/theorems/632) a tautology rather than a deep theorem connecting two independently meaningful operations. The integral should have its own definition — rooted in approximation and area — so that its relationship to [differentiation](/page/Derivative) becomes a genuine mathematical discovery. ### Riemann's Insight: Approximation by Rectangles Riemann's resolution is to approximate the area from above and below simultaneously. Partition $[a, b]$ into subintervals and construct two families of rectangles: tall ones that overshoot $f$ on each subinterval (using the supremum of $f$) and short ones that undershoot it (using the infimum). The overshoot gives an upper bound for the area, the undershoot gives a lower bound, and the true area — if it exists — is trapped between the two. As the partition is refined, both families of rectangles fit the curve more tightly. If the upper and lower bounds can be driven together to a common limit, that limit is the integral of $f$. This definition is intrinsic to $f$: it requires no antiderivative, it works for any bounded function for which the upper and lower bounds converge, and it reduces the existence of the integral to the single quantitative condition that the oscillation of $f$ on small intervals is controllable. [/motivation] ## Definition The construction proceeds in three stages: first the combinatorial notion of a partition, then the upper and lower sums that approximate the integral, and finally the passage to the [supremum and infimum](/page/Supremum%20and%20Infimum) over all partitions. [definition:Partition] Let $a, b \in \mathbb{R}$ with $a < b$. A **partition** of $[a, b]$ is a finite set $P = \{x_0, x_1, \ldots, x_n\}$ satisfying \begin{align*} a = x_0 < x_1 < x_2 < \cdots < x_n = b. \end{align*} The **mesh** (or **norm**) of $P$ is \begin{align*} \|P\| := \max_{1 \le i \le n} (x_i - x_{i-1}). \end{align*} A partition $Q$ is a **refinement** of $P$ if $P \subseteq Q$ as [sets](/page/Set). [/definition] Every partition decomposes $[a, b]$ into $n$ non-overlapping subintervals $[x_{i-1}, x_i]$ of widths $\Delta x_i := x_i - x_{i-1}$. Refinement only adds points, so it splits existing subintervals into smaller pieces without removing any. The key idea is to estimate the contribution of $f$ on each subinterval by its extreme values. Since $f$ is bounded, the supremum and infimum on each subinterval are finite, and the resulting sums bracket the true area. [definition:Upper And Lower Darboux Sums] Let $f: [a, b] \to \mathbb{R}$ be a bounded function and $P = \{x_0, \ldots, x_n\}$ a partition of $[a, b]$. For each $i \in \{1, \ldots, n\}$, define \begin{align*} M_i &:= \sup_{x \in [x_{i-1}, x_i]} f(x), \\ m_i &:= \inf_{x \in [x_{i-1}, x_i]} f(x). \end{align*} The **upper Darboux sum** and **lower Darboux sum** of $f$ with respect to $P$ are \begin{align*} U(f, P) &:= \sum_{i=1}^n M_i \, \Delta x_i, \\ L(f, P) &:= \sum_{i=1}^n m_i \, \Delta x_i. \end{align*} [/definition] For any partition $P$, the inequality $m_i \le M_i$ on each subinterval gives $L(f, P) \le U(f, P)$. More importantly, refinement improves the approximation: if $Q$ is a refinement of $P$, then $L(f, P) \le L(f, Q)$ and $U(f, Q) \le U(f, P)$, because splitting a subinterval can only raise the infimum and lower the supremum. This monotonicity is what makes the following definition possible. [definition:Upper And Lower Darboux Integrals] Let $f: [a, b] \to \mathbb{R}$ be bounded. The **upper Darboux integral** and **lower Darboux integral** of $f$ are \begin{align*} \overline{\int_a^b} f(x)\, dx &:= \inf_P \, U(f, P), \\ \underline{\int_a^b} f(x)\, dx &:= \sup_P \, L(f, P), \end{align*} where the infimum and supremum are taken over all partitions $P$ of $[a, b]$. [/definition] The upper integral squeezes down from above by taking the tightest possible overshoot, and the lower integral pushes up from below by taking the largest possible undershoot. For any two partitions $P$ and $Q$, their common refinement $P \cup Q$ satisfies $L(f, P) \le L(f, P \cup Q) \le U(f, P \cup Q) \le U(f, Q)$, so every lower sum is at most every upper sum. Taking the supremum over $P$ on the left and the infimum over $Q$ on the right yields the fundamental inequality \begin{align*} \underline{\int_a^b} f(x)\, dx \le \overline{\int_a^b} f(x)\, dx. \end{align*} The two integrals need not be equal — but when they are, the function is integrable. [definition:Riemann Integrability] A bounded function $f: [a, b] \to \mathbb{R}$ is **Riemann integrable** (or **Darboux integrable**) if \begin{align*} \underline{\int_a^b} f(x)\, dx = \overline{\int_a^b} f(x)\, dx. \end{align*} The common value is the **Riemann integral** of $f$ over $[a, b]$, denoted \begin{align*} \int_a^b f(x)\, dx. \end{align*} The set of all Riemann integrable functions on $[a, b]$ is denoted $\mathcal{R}([a, b])$. [/definition] [remark:Riemann Sums Versus Darboux Sums] There is a second, historically prior, formulation: a **Riemann sum** chooses sample points $t_i \in [x_{i-1}, x_i]$ and forms $S(f, P, t) = \sum_{i=1}^n f(t_i) \, \Delta x_i$. One declares $f$ integrable if these sums converge to a common limit as $\|P\| \to 0$, regardless of the choice of sample points. The Darboux formulation (via upper and lower sums) and the Riemann formulation are equivalent — the Darboux approach is technically cleaner because it avoids quantifying over sample points, but the Riemann sum formulation is indispensable for numerical computation and for recognising limits as integrals. [/remark] ## The Darboux Criterion The definition of Riemann integrability asks for the upper and lower integrals to coincide, but checking this directly requires computing two global extrema over all partitions. A much more useful characterisation reduces integrability to a single quantitative condition: for every $\varepsilon > 0$, there exists a partition making the gap between the upper and lower sums smaller than $\varepsilon$. This reformulation is the workhorse behind every integrability proof. [quotetheorem:281] The forward direction is immediate from the definitions: if $\overline{\int} f = \underline{\int} f$, then for any $\varepsilon > 0$ one can find partitions $P_1, P_2$ with $U(f, P_1) < \overline{\int} f + \varepsilon/2$ and $L(f, P_2) > \underline{\int} f - \varepsilon/2$, and their common refinement $P = P_1 \cup P_2$ satisfies $U(f, P) - L(f, P) < \varepsilon$. The reverse direction is equally direct: if $U(f, P) - L(f, P) < \varepsilon$ for some $P$, then $\overline{\int} f - \underline{\int} f \le U(f, P) - L(f, P) < \varepsilon$, and since $\varepsilon$ is arbitrary, the two integrals coincide. The power of this criterion is that it converts an existential problem (does the integral exist?) into a constructive one (can you find a good enough partition?). It also makes the connection to oscillation transparent: the difference $U(f, P) - L(f, P) = \sum_{i=1}^n (M_i - m_i) \Delta x_i$ is precisely the sum of the oscillations of $f$ on each subinterval, weighted by the subinterval length. Integrability therefore means that the total weighted oscillation can be made arbitrarily small. ### The Oscillation Function The observation above motivates isolating the oscillation as a function in its own right, which will be essential for the Lebesgue criterion. [definition:Oscillation Of A Function] Let $f: [a, b] \to \mathbb{R}$ be bounded. The **oscillation** of $f$ on a set $S \subseteq [a, b]$ is \begin{align*} \omega_f(S) := \sup_{x, y \in S} |f(x) - f(y)| = \sup_S f - \inf_S f. \end{align*} The **oscillation of $f$ at a point** $x_0 \in [a, b]$ is \begin{align*} \omega_f(x_0) := \lim_{\delta \to 0^+} \omega_f\big([a, b] \cap (x_0 - \delta, x_0 + \delta)\big) = \inf_{\delta > 0} \omega_f\big([a, b] \cap (x_0 - \delta, x_0 + \delta)\big). \end{align*} [/definition] The oscillation $\omega_f(x_0)$ measures how badly $f$ fails to be continuous at $x_0$: the function $f$ is continuous at $x_0$ if and only if $\omega_f(x_0) = 0$. The infimum in the definition is actually a limit because $\omega_f([a,b] \cap (x_0 - \delta, x_0 + \delta))$ is non-increasing as $\delta$ decreases. ## Integrability Criteria Which bounded functions are Riemann integrable? The Darboux criterion reduces this question to controlling oscillation, and the three major integrability results correspond to three increasingly general ways of doing so. ### Continuous Functions The simplest class of Riemann integrable functions is the continuous functions. [Continuity](/page/Continuity) on a compact interval is uniform, and uniform continuity is exactly the condition needed to control the oscillation simultaneously across all subintervals. [quotetheorem:282] The proof is a direct application of the Darboux criterion and uniform continuity. Since $[a, b]$ is compact and $f$ is continuous, $f$ is uniformly continuous: for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|f(x) - f(y)| < \varepsilon / (b - a)$ whenever $|x - y| < \delta$. Choosing any partition $P$ with $\|P\| < \delta$ ensures that $M_i - m_i < \varepsilon / (b - a)$ on each subinterval, so \begin{align*} U(f, P) - L(f, P) = \sum_{i=1}^n (M_i - m_i) \Delta x_i < \frac{\varepsilon}{b - a} \sum_{i=1}^n \Delta x_i = \varepsilon. \end{align*} The hypothesis of compactness is essential: the function $f: (0, 1] \to \mathbb{R}$ defined by $f(x) = \sin(1/x)$ is continuous but the interval is not closed, and extending $f$ to $[0, 1]$ by any value at $0$ produces a bounded function that is still Riemann integrable — but this requires the Lebesgue criterion, not uniform continuity alone. ### Monotone Functions Monotone functions need not be continuous (they can have countably many jump discontinuities), yet they are always Riemann integrable. The key is that monotonicity forces the oscillation on each subinterval to telescope. [quotetheorem:194] Suppose $f$ is non-decreasing (the non-increasing case follows by considering $-f$). Then $M_i = f(x_i)$ and $m_i = f(x_{i-1})$ for each $i$. For a uniform partition with $n$ subintervals, $\Delta x_i = (b - a)/n$ for all $i$, and the Darboux sum difference telescopes: \begin{align*} U(f, P) - L(f, P) &= \sum_{i=1}^n \big(f(x_i) - f(x_{i-1})\big) \frac{b - a}{n} = \frac{b - a}{n} \big(f(b) - f(a)\big). \end{align*} Given $\varepsilon > 0$, choosing $n > (b - a)(f(b) - f(a)) / \varepsilon$ makes the difference less than $\varepsilon$, and the Darboux criterion is satisfied. The telescoping mechanism here is specific to monotone functions: it converts the sum of oscillations into a single boundary difference. For non-monotone functions, no such cancellation occurs, and integrability depends on a more delicate analysis of the discontinuity set. ### The Lebesgue Criterion The definitive answer to "which bounded functions are Riemann integrable?" is the Lebesgue criterion, which characterises integrability in terms of the size of the discontinuity set. A bounded function is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero — that is, the discontinuities, while possibly uncountable, can be covered by intervals of arbitrarily small total length. [definition:Lebesgue Measure Zero] A set $E \subseteq \mathbb{R}$ has **Lebesgue measure zero** (or is a **null set**) if for every $\varepsilon > 0$ there exists a countable collection of open intervals $\{(a_k, b_k)\}_{k=1}^\infty$ such that \begin{align*} E \subseteq \bigcup_{k=1}^\infty (a_k, b_k) \quad \text{and} \quad \sum_{k=1}^\infty (b_k - a_k) < \varepsilon. \end{align*} [/definition] [Countable sets](/page/Countable%20Set) are null (cover the $k$-th point by an interval of length $\varepsilon / 2^k$), but null sets can be uncountable — the Cantor set is uncountable with Lebesgue measure zero. The notion of "measure zero" is the minimal amount of measure theory needed to state the Lebesgue criterion, and it is a precursor to the full Lebesgue measure $\mathcal{L}^1$ on $\mathbb{R}$. [theorem:Lebesgue Criterion For Riemann Integrability] Let $f: [a, b] \to \mathbb{R}$ be bounded, and let $D_f := \{x \in [a, b] : f \text{ is discontinuous at } x\}$. Then $f \in \mathcal{R}([a, b])$ if and only if $D_f$ has Lebesgue measure zero. [/theorem] This theorem subsumes both of the preceding results: continuous functions have $D_f = \varnothing$ (measure zero), and monotone functions have $D_f$ countable (hence measure zero, since each jump discontinuity is an isolated point in the range). The Lebesgue criterion also explains precisely why the Dirichlet function $\mathbb{1}_\mathbb{Q} : [0, 1] \to \mathbb{R}$ is not Riemann integrable: it is discontinuous at every point, and $D_f = [0, 1]$ has measure $1$. More subtly, the criterion shows that the indicator function of the Cantor set $\mathbb{1}_C$ is Riemann integrable — despite being discontinuous at uncountably many points — because $D_f = C$ has measure zero. The proof of the Lebesgue criterion proceeds by connecting the Darboux criterion to the oscillation function. The difference $U(f, P) - L(f, P) = \sum_{i=1}^n \omega_f([x_{i-1}, x_i]) \, \Delta x_i$ is small if and only if the subintervals where $\omega_f$ is large have small total length. The set $\{x : \omega_f(x) \ge \alpha\}$ is closed for each $\alpha > 0$ (the oscillation function is upper semicontinuous), and the discontinuity set decomposes as $D_f = \bigcup_{k=1}^\infty \{x : \omega_f(x) \ge 1/k\}$. Integrability therefore amounts to each of these [closed sets](/page/Closed%20Set) being coverable by intervals of small total length — which is exactly the measure-zero condition on $D_f$. [example:The Thomae Function] The **Thomae function** (or "popcorn function") $T: [0, 1] \to \mathbb{R}$ is defined by \begin{align*} T: [0, 1] &\to \mathbb{R} \\ x &\mapsto \begin{cases} 1/q & \text{if } x = p/q \text{ in lowest terms with } q \ge 1, \\ 0 & \text{if } x \notin \mathbb{Q}. \end{cases} \end{align*} This function is discontinuous at every rational (since irrationals are dense, every neighbourhood of a rational contains points where $T = 0$, but $T(p/q) = 1/q > 0$) and continuous at every irrational (since for any $\varepsilon > 0$, there are only finitely many rationals $p/q$ in $[0, 1]$ with $q \le 1/\varepsilon$, so a sufficiently small neighbourhood of an irrational misses all of them). The discontinuity set is $D_T = \mathbb{Q} \cap [0, 1]$, which is countable and hence has Lebesgue measure zero. By the Lebesgue criterion, $T \in \mathcal{R}([0, 1])$, and since $T(x) = 0$ for all $x \in [0, 1] \setminus \mathbb{Q}$ and the rationals form a null set, the integral is \begin{align*} \int_0^1 T(x)\, dx = 0. \end{align*} This is a striking example: a function that is discontinuous on a dense set, yet Riemann integrable with integral zero. [/example] [example:Failure Of The Dirichlet Function] The **Dirichlet function** $D: [0, 1] \to \mathbb{R}$ is defined by \begin{align*} D: [0, 1] &\to \mathbb{R} \\ x &\mapsto \begin{cases} 1 & \text{if } x \in \mathbb{Q}, \\ 0 & \text{if } x \notin \mathbb{Q}. \end{cases} \end{align*} On every subinterval $[x_{i-1}, x_i]$ of every partition, the density of both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ ensures that $M_i = 1$ and $m_i = 0$. Therefore \begin{align*} U(D, P) = \sum_{i=1}^n 1 \cdot \Delta x_i = 1, \qquad L(D, P) = \sum_{i=1}^n 0 \cdot \Delta x_i = 0 \end{align*} for every partition $P$. The upper and lower integrals are $\overline{\int_0^1} D = 1$ and $\underline{\int_0^1} D = 0$, so $D \notin \mathcal{R}([0, 1])$. The Lebesgue criterion confirms this: $D$ is discontinuous everywhere, and $D_D = [0, 1]$ has measure $1$. The Dirichlet function is, however, Lebesgue integrable with $\int_{[0,1]} D \, d\mathcal{L}^1 = 0$, since $\mathbb{Q}$ has measure zero. [/example] ## Properties of the Riemann Integral The Riemann integral inherits algebraic and order properties directly from the structure of the Darboux sums. These properties are used throughout calculus and analysis, and their proofs all follow the same pattern: establish the corresponding inequality for Darboux sums, then pass to the limit over partitions. The fundamental properties — linearity, monotonicity, the triangle inequality, and closure under products — are collected in the following theorem. [quotetheorem:209] These four properties make $\mathcal{R}([a, b])$ a real algebra and the integral a positive linear functional on it. Each deserves individual discussion. ### Linearity Linearity (property (i)) is the most fundamental algebraic property. The proof reduces to the fact that $\sup(f + g) \le \sup f + \sup g$ on each subinterval, which gives $U(f + g, P) \le U(f, P) + U(g, P)$, with equality when one of the functions is constant on each subinterval. The scalar multiplication case $U(\lambda f, P) = \lambda \, U(f, P)$ (for $\lambda \ge 0$) is immediate. Combined, these give integrability of $\lambda f + g$ and the identity $\int (\lambda f + g) = \lambda \int f + \int g$. ### Monotonicity and the Triangle Inequality Monotonicity (property (ii)) states that the integral preserves order. The proof is direct: $f \le g$ implies $m_i^f \le m_i^g$ on each subinterval, so $L(f, P) \le L(g, P)$ for every partition. Taking the supremum gives $\underline{\int} f \le \underline{\int} g$, and since both functions are integrable, the lower integrals equal the integrals. Property (iii) — that $|f|$ is integrable with $|\int f| \le \int |f|$ — follows from monotonicity applied to $-|f| \le f \le |f|$. The integrability of $|f|$ itself uses the fact that the absolute value function is Lipschitz, so the [Integrability of Continuous Composition](/theorems/284) theorem applies. ### Additivity Over Intervals The integral is additive over adjacent intervals: splitting the domain of integration at a point inside $[a, b]$ decomposes the integral into two pieces. [theorem:Additivity Over Intervals] Let $f \in \mathcal{R}([a, b])$ and $c \in (a, b)$. Then $f \in \mathcal{R}([a, c])$ and $f \in \mathcal{R}([c, b])$, and \begin{align*} \int_a^b f(x)\, dx = \int_a^c f(x)\, dx + \int_c^b f(x)\, dx. \end{align*} Conversely, if $f \in \mathcal{R}([a, c])$ and $f \in \mathcal{R}([c, b])$, then $f \in \mathcal{R}([a, b])$. [/theorem] The proof uses the Darboux criterion on each piece. Every partition of $[a, b]$ can be refined to include the point $c$ without increasing the Darboux sum difference, and a partition containing $c$ decomposes into a partition of $[a, c]$ and a partition of $[c, b]$. This property extends by induction to any finite decomposition of $[a, b]$ into subintervals and is essential for defining improper integrals, where one splits $[a, b]$ at a singular point and takes limits from each side. ## The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus is the central result connecting the two operations of differentiation and integration. It comes in two parts: the first says that integration followed by differentiation recovers the original function; the second says that differentiation followed by integration recovers the function up to a boundary evaluation. These are *not* formal inverses — each requires hypotheses the other does not — and understanding precisely where each part applies is essential. ### The Integral as an Antiderivative Given a Riemann integrable function, one can always form the "running integral" $F(x) = \int_a^x f(t)\, dt$. This function is automatically continuous (indeed Lipschitz). The question is whether $F$ is differentiable and, if so, whether $F' = f$. The answer is yes at every point where $f$ is continuous, but the conclusion can fail at discontinuities of $f$. [quotetheorem:190] The continuity of $F$ follows from the estimate $|F(x) - F(y)| \le M |x - y|$ where $M = \sup_{[a,b]} |f|$, which makes $F$ Lipschitz. For differentiability at $c$, one writes \begin{align*} \frac{F(c + h) - F(c)}{h} - f(c) = \frac{1}{h} \int_c^{c+h} \big(f(t) - f(c)\big)\, dt, \end{align*} and the continuity of $f$ at $c$ ensures that $|f(t) - f(c)| < \varepsilon$ for $|t - c|$ small, making the right-hand side less than $\varepsilon$ in absolute value. The hypothesis that $f$ is continuous at $c$ cannot be dropped: if $f$ has a jump discontinuity at $c$, then $F$ has a corner there (it is Lipschitz but not differentiable). ### Evaluation by Antiderivatives The second part runs in the opposite direction: if a function already has a continuous derivative, then its integral can be computed by boundary evaluation. This is the theorem that powers all explicit integral computations in calculus. [quotetheorem:191] The proof uses the [Mean Value Theorem](/theorems/186) on each subinterval of a partition: for each $i$, there exists $c_i \in (x_{i-1}, x_i)$ with $G(x_i) - G(x_{i-1}) = G'(c_i) \, \Delta x_i = f(c_i) \, \Delta x_i$. Summing over $i$ telescopes to $G(b) - G(a) = \sum_{i=1}^n f(c_i) \, \Delta x_i$, which is a Riemann sum for $f$. Since $f$ is continuous on $[a, b]$, it is Riemann integrable (by the [Integrability of Continuous Functions](/theorems/282)), and this Riemann sum converges to $\int_a^b f(x)\, dx$ as $\|P\| \to 0$. The continuity hypothesis on $f$ can be weakened: the result extends to the case where $G$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $G' \in \mathcal{R}([a, b])$ — but this more general statement requires a separate argument. Even so, the hypothesis that the derivative is Riemann integrable is necessary: the Volterra function provides a differentiable function whose derivative is bounded but not Riemann integrable (discontinuous on a set of positive measure). This limitation is resolved by the [Lebesgue integral](/page/Lebesgue%20Integral), where every bounded derivative is integrable. [example:The Integral Of A Power Function] Let $n \in \mathbb{N}$ and consider $f: [0, 1] \to \mathbb{R}$ defined by $f(x) = x^n$. The function $F: [0, 1] \to \mathbb{R}$ defined by $F(x) = x^{n+1}/(n+1)$ satisfies $F'(x) = x^n = f(x)$ for all $x \in [0, 1]$, and $F$ is continuous on $[0, 1]$ with $F' = f$ continuous (hence Riemann integrable). By the Second Fundamental Theorem: \begin{align*} \int_0^1 x^n \, dx = F(1) - F(0) = \frac{1}{n+1} - 0 = \frac{1}{n+1}. \end{align*} This computation can also be verified directly from the definition using the identity $\sum_{i=1}^N i^n = N^{n+1}/(n+1) + O(N^n)$ and a uniform partition, confirming that the two approaches agree. [/example] ## Limitations and the Passage to Lebesgue The Riemann integral is adequate for continuous and piecewise-continuous functions on bounded intervals, but it has structural deficiencies that become serious in analysis. Understanding these limitations clarifies why the Lebesgue integral is necessary and what it corrects. ### Failure of Completeness The space $\mathcal{R}([a, b])$ with the norm $\|f\| = \sup_{[a,b]} |f|$ is not complete — there exist [Cauchy sequences](/page/Cauchy%20Sequence) of Riemann integrable functions whose pointwise limits are not Riemann integrable. A monotone sequence of Riemann integrable functions can converge to the Dirichlet function, which is not in $\mathcal{R}([0, 1])$. This is a fundamental deficiency: one cannot do analysis in a space that is not complete, because limits of approximation schemes may escape the space. [example:A Non Integrable Pointwise Limit] Let $\{q_1, q_2, q_3, \ldots\}$ be an enumeration of $\mathbb{Q} \cap [0, 1]$. Define $f_n: [0, 1] \to \mathbb{R}$ by \begin{align*} f_n: [0, 1] &\to \mathbb{R} \\ x &\mapsto \begin{cases} 1 & \text{if } x \in \{q_1, \ldots, q_n\}, \\ 0 & \text{otherwise}. \end{cases} \end{align*} Each $f_n$ is zero except at finitely many points, so $f_n \in \mathcal{R}([0, 1])$ with $\int_0^1 f_n \, dx = 0$. The sequence $f_n$ is non-decreasing and converges pointwise to the Dirichlet function $D = \mathbb{1}_\mathbb{Q}$, which satisfies $D \notin \mathcal{R}([0, 1])$. The pointwise limit of Riemann integrable functions need not be Riemann integrable — and even when it is, the limit of the integrals need not equal the integral of the limit without additional hypotheses. [/example] ### Convergence Theorems The Riemann integral has limited convergence theorems. The strongest positive result is that [uniform convergence](/page/Uniform%20Convergence) preserves Riemann integrability and commutes with integration. [quotetheorem:259] Uniform convergence is a strong hypothesis — it requires the convergence to be equally fast at every point, which rules out many natural sequences arising in Fourier analysis, probability, and PDE. The Lebesgue theory provides the [Monotone Convergence Theorem](/theorems/509) and the [Dominated Convergence Theorem](/theorems/4), which allow pointwise convergence under much weaker conditions (monotonicity or domination by an integrable function). These convergence theorems are the primary technical advantage of the Lebesgue integral over the Riemann integral and are indispensable in modern analysis. ### Why Lebesgue Succeeds The Lebesgue integral resolves these deficiencies by partitioning the *range* of $f$ rather than its domain. Instead of asking "what does $f$ do on each subinterval?", the Lebesgue approach asks "on what set does $f$ take values in a given range?" — and measures the size of that set using Lebesgue measure $\mathcal{L}^1$. This change of perspective has three consequences. First, the class of integrable functions is strictly larger: every Riemann integrable function is Lebesgue integrable (with the same value), but the Dirichlet function and many other bounded functions become integrable. Second, the space $L^1([a, b], \mathcal{L}^1)$ is complete — limits of Cauchy sequences remain in the space. Third, the powerful convergence theorems (Monotone Convergence, Dominated Convergence, Fatou's Lemma) hold for Lebesgue integrals, making the theory robust under limiting operations. ## References - Rudin, *Principles of Mathematical Analysis* (1976). - Apostol, *Mathematical Analysis* (1974). - Bartle and Sherbert, *Introduction to Real Analysis* (2011). - Tao, *Analysis I* (2016).