The central question of this course is deceptively simple: when can we interchange a limit and an integral? That is, when does

\begin{align*} \lim_{n \to \infty} \int f_n \, d\mu = \int \lim_{n \to \infty} f_n \, d\mu \end{align*}

hold? In the Riemann theory taught in Analysis II, the answer is unsatisfying: only under strong hypotheses (typically [uniform convergence](/page/Uniform%20Convergence)) that exclude many natural situations in analysis and probability. Measure theory was created, in large part, to give a much better answer.

# Motivation

## Why Riemann Integration Breaks Down

The [Riemann integral](/page/Riemann%20Integral) partitions the *domain* of a function into subintervals, forms upper and lower sums, and declares the function integrable when these sums converge to the same limit. This works beautifully for continuous [functions](/page/Function), and even for functions with "few" discontinuities — the precise condition is that the set of discontinuities has Lebesgue measure zero, though the Riemann theory cannot even state this condition in its own language. The trouble begins when we try to take [limits](/page/Limit).

Consider the following example. Enumerate the rationals in $[0,1]$ as $q_1, q_2, q_3, \dots$ and define $f_n = \mathbf{1}_{\{q_1, \dots, q_n\}}$. Each $f_n$ is zero except at finitely many points, so each is Riemann integrable with $\int_0^1 f_n(x)\,dx = 0$. The pointwise limit $f = \lim_n f_n = \mathbf{1}_{\mathbb{Q} \cap [0,1]}$ exists everywhere — yet $f$ is *not* Riemann integrable. Every subinterval of $[0,1]$ contains both rationals (where $f = 1$) and irrationals (where $f = 0$), so the upper and lower Riemann sums are permanently stuck at $1$ and $0$ respectively. The Riemann integral cannot pass the limit through the integral sign, even though the sequence is monotone increasing and uniformly bounded.

This is not an isolated pathology. The three situations where the interchange of limits and integrals matters most are precisely where the Riemann theory is weakest:

**Monotone limits.** If $0 \leq f_1 \leq f_2 \leq \cdots$ pointwise, can we conclude $\int \lim f_n = \lim \int f_n$? Not in the Riemann theory — the limit function may fail to be Riemann integrable, as the example above shows.

**Limits with a dominating function.** If $f_n \to f$ pointwise and $|f_n| \leq g$ for some integrable $g$, can we interchange the limit and integral? Riemann theory requires the much stronger hypothesis of uniform convergence, which rules out many natural examples — for instance, approximating a discontinuous function by continuous ones.

**One-sided bounds on integrals of limits.** If $f_n \geq 0$ pointwise, what can we say about $\int \liminf f_n$ relative to $\liminf \int f_n$? The Riemann theory has no result of this kind at all.

## How Measure Theory Resolves These Failures

The [Lebesgue integral](/page/Lebesgue%20Integral), built on the theory of measures, resolves every one of these difficulties. The key insight is to partition the *range* of a function rather than its domain: instead of asking "how wide is each subinterval?", we ask "how large is the set where $f$ takes values in a given range?". This requires a systematic notion of "size" for subsets — a *measure* — and in particular the ability to assign size zero to complicated [sets](/page/Set) like $\mathbb{Q} \cap [0,1]$. The payoff is a suite of convergence theorems that the Riemann theory cannot match.

The following table summarises the core exchange. Each row identifies a limitation of the Riemann integral and the measure-theoretic result that overcomes it.

| **Problem in Riemann theory** | **Measure-theoretic resolution** | |---|---| | Monotone limits of Riemann-integrable functions may have non-Riemann-integrable limits, so $\lim \int f_n$ and $\int \lim f_n$ cannot be compared. | The [Monotone Convergence Theorem](/theorems/509) guarantees $\int \lim f_n = \lim \int f_n$ for any increasing sequence of non-negative measurable functions — no [integrability](/page/Integral) assumption on the limit is needed; the theorem *produces* it. | | Interchanging limits and integrals requires uniform convergence, which is far too restrictive for most applications. | The [Dominated Convergence Theorem](/theorems/4) requires only pointwise convergence and a single integrable dominating function $g$ with $\lvert f_n \rvert \leq g$. Uniform convergence is replaced by a pointwise bound. | | No Riemann analogue exists for bounding $\int \liminf f_n$ when the sequence is not monotone. | [Fatou's Lemma](/theorems/510) gives the one-sided inequality $\int \liminf f_n \leq \liminf \int f_n$ for any sequence of non-negative measurable functions — a result with no Riemann counterpart. | | The class of Riemann-integrable functions is not closed under pointwise limits, so the theory is not self-contained. | The class of Lebesgue-integrable functions is closed under pointwise limits (with an integrable dominator), making the Lebesgue theory internally consistent and stable under the operations of analysis. | | Riemann integration is defined only on intervals in $\mathbb{R}^n$ and cannot handle abstract domains. | Lebesgue integration is defined on arbitrary measure spaces $(X, \mathcal{A}, \mu)$, including probability spaces, manifolds, and function spaces. |

The three convergence theorems — MCT, Fatou, DCT — are the engine of the entire course. Every major result we prove will rely on at least one of them, often in combination.

## Why Probability Needs Measure Theory

The connection to probability is immediate: a probability space is simply a measure space $(X, \mathcal{A}, \mu)$ with the additional constraint that $\mu(X) = 1$. A random variable is a measurable function, and its expectation is a Lebesgue integral. Once this identification is made, the convergence theorems become theorems about expectations of random variables — and the limitations of the Riemann integral become limitations on what can be proved about random phenomena.

The fundamental questions of probability theory concern the long-run behaviour of [sequences](/page/Sequence) of random variables: does the sample average $\bar{X}_n = (X_1 + \cdots + X_n)/n$ converge to the population mean? In what sense? How are the fluctuations distributed? Answering these questions requires precisely the tools that the Riemann integral lacks. The following table shows how the measure-theoretic machinery developed in the first half of the course underpins the probabilistic results of the second half.

| **Probabilistic question** | **Measure-theoretic tool** | |---|---| | Does $\bar{X}_n \to \mu$ almost surely for i.i.d. random variables with $\mathbb{E}[\lvert X_1 \rvert] < \infty$? | The [Strong Law of Large Numbers](/theorems/520), proved via the [Birkhoff Ergodic Theorem](/theorems/518), which itself is established using the Maximal Ergodic Theorem and the Dominated Convergence Theorem. | | What is the limiting distribution of the standardised partial sums $(S_n - n\mu)/(\sigma\sqrt{n})$? | The [Central Limit Theorem](/theorems/521), proved using characteristic functions (Fourier transforms of probability measures) and Lévy's convergence theorem. | | When can "infinitely many" events occur? When must they? | The [Borel–Cantelli Lemmas](/theorems/507): the first uses countable subadditivity of measures; the second uses independence to convert a divergent series into an almost-sure event. | | When can expectations and limits be interchanged for random variables? | The [Dominated Convergence Theorem](/theorems/4) applied in the probability setting, often combined with uniform integrability. | | How do time averages relate to space averages for stationary processes? | The [Birkhoff Ergodic Theorem](/theorems/518), which asserts $n^{-1}\sum_{k=0}^{n-1} f \circ \Theta^k \to \bar{f}$ almost everywhere for measure-preserving transformations $\Theta$. |