The central problem of partial differential equations is existence: given a PDE such as $\Delta f = g$, does a solution exist, and in what sense? Classical methods require candidates to be twice continuously differentiable, but the spaces $C^k(\mathbb{R}^n)$ have terrible compactness properties — the closed unit ball is never compact in infinite dimensions, and [sequences](/page/Sequence) of smooth [functions](/page/Function) can converge pointwise to discontinuous [limits](/page/Limit). This makes it impossible to extract convergent subsequences from approximating sequences, which is the standard strategy for proving existence.

The resolution is to work in larger, better-behaved function spaces — the $L^p$ spaces and their cousins the [Sobolev spaces](/page/Sobolev%20Space) $W^{k,p}$ — where completeness and compactness are available. The cost is that functions in these spaces are only defined up to [sets](/page/Set) of measure zero, and "derivatives" must be reinterpreted in a weak ([distributional](/page/Distribution)) sense. The payoff is enormous: the weak formulation of a PDE makes existence a consequence of functional analysis (Riesz representation, Lax-Milgram), and the Sobolev embedding theorems then bootstrap the weak solution back to classical regularity.

This course develops the machinery needed for this program. Chapter 1 establishes the properties of $L^p$ spaces — completeness, density, [separability](/page/Separable), and regularity results (Lebesgue [differentiation](/page/Derivative), Lusin's theorem) — that form the analytic foundation. Chapter 2 addresses the "hunt for compactness" in infinite-dimensional spaces: weak and weak-$*$ topologies, reflexivity (Kakutani), uniform convexity (Milman-Pettis), and the identification of dual spaces. Chapter 3 develops Fourier analysis as a tool for decomposing functions and reducing PDEs to algebraic equations. Chapter 4 introduces Sobolev spaces, proves the Sobolev embedding theorems, and applies everything to the Dirichlet problem for Poisson's equation.

# Integration

Measure theory and Lebesgue integration provide the technical substrate on which the entire course is built. The key advantage of the [Lebesgue integral](/page/Lebesgue%20Integral) over the [Riemann integral](/page/Riemann%20Integral) is not greater generality per se, but the availability of powerful limit theorems — the Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem — that allow us to interchange limits and integrals under controlled hypotheses. These theorems fail for the Riemann integral: if $\{q_1, q_2, \ldots\}$ is an enumeration of $\mathbb{Q} \cap [0,1]$, the sequence $f_n := \mathbb{1}_{\{q_1, \ldots, q_n\}}$ converges pointwise to $\mathbb{1}_{\mathbb{Q} \cap [0,1]}$, each $f_n$ is Riemann integrable, but the limit is not. The Lebesgue theory handles this effortlessly.

## [Integrability](/page/Integral) and the Convergence Theorems

The Lebesgue integral is constructed in three stages: first for simple functions (measurable functions taking finitely many values), then for non-negative measurable functions (via supremum over simple functions below), and finally for general measurable functions (by decomposing into positive and negative parts). For a simple function $s = \sum_{i=1}^n \alpha_i \mathbb{1}_{A_i}$ on a measure space $(E, \mathcal{E}, \mu)$, the integral over a measurable set $A$ is $\int_A s \, d\mu := \sum_{i=1}^n \alpha_i \mu(A_i \cap A)$. For a non-negative measurable function $f$, $\int_E f \, d\mu := \sup\{\int_E s \, d\mu : 0 \le s \le f, \, s \text{ simple}\}$.

A basic but useful estimate relating the integral to the measure of level sets is the Markov (or Chebyshev) inequality: if $f \ge 0$ is measurable and $\alpha > 0$, then $\mu(\{f \ge \alpha\}) \le \frac{1}{\alpha} \int f \, d\mu$. This follows immediately by noting that $\alpha \mathbb{1}_{\{f \ge \alpha\}} \le f$.

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A direct consequence is that if $\int_E f \, d\mu < \infty$ for $f \ge 0$, then $f < \infty$ a.e.: the sets $A_n := \{f \ge n\}$ decrease to $\{f = +\infty\}$, and $\mu(A_n) \le \frac{1}{n} \int f \, d\mu \to 0$.

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The three convergence theorems are the workhorses of the Lebesgue theory. The Monotone Convergence Theorem handles the simplest case: non-negative functions increasing to a limit.

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The proof constructs, for each simple function $s \le f$ and each $c \in (0,1)$, the sets $E_k := \{x : f_k(x) \ge c \cdot s(x)\}$, which increase to $E$ by pointwise convergence. The key step is that the map $A \mapsto \int_A s \, d\mu$ is itself a measure (by the $\sigma$-additivity of $\mu$ and the finiteness of the sum defining $s$), so [continuity](/page/Continuity) from below gives $\int_{E_k} s \, d\mu \uparrow \int_E s \, d\mu$. The inequality $\int f_k \, d\mu \ge c \int_{E_k} s \, d\mu$ (valid since $f_k \ge cs$ on $E_k$) then yields $\lim \int f_k \, d\mu \ge c \int s \, d\mu$, and taking $c \uparrow 1$ and the supremum over $s$ finishes the argument. The non-negativity and monotonicity hypotheses are both essential: without monotonicity, mass can escape to infinity (consider $f_n = n \mathbb{1}_{(0, 1/n]}$ on $[0,1]$, where $\int f_n = 1$ for all $n$ but $f_n \to 0$ pointwise).

An immediate and important corollary is that for non-negative measurable functions $(f_k)_k$, $\int \sum_k f_k \, d\mu = \sum_k \int f_k \, d\mu$ — infinite sums and integrals commute when everything is non-negative.

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When the sequence is not monotone, we lose equality but retain a one-sided estimate.

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Fatou's Lemma is proved by applying the Monotone Convergence Theorem to the increasing sequence $g_n := \inf_{k \ge n} f_k$, which satisfies $g_n \uparrow \liminf_k f_k$ and $g_n \le f_k$ for all $k \ge n$. The inequality can be strict: the sliding bump $f_n = n \mathbb{1}_{(0,1/n]}$ has $\liminf f_n = 0$ a.e. but $\liminf \int f_n = 1$.

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