Summary of Key Results

| Theorem | Conditions | Conclusion | | ---------------------- | ----------------------------------------- | ---------------------------------------- | --------------- | ------------------------ | | Monotone Convergence | $0 \leq f_n \nearrow f$ | $\lim \int f_n = \int f$ | | Fatou”s Lemma | $f_n \geq 0$ | $\int \liminf f_n \leq \liminf \int f_n$ | | Dominated Convergence | $f_n \to f$, $| f_n | \leq g \in L^1$ | $\lim \int f_n = \int f$ | | Holder’s Inequality | $f \in L^p$, $g \in L^q$, $1/p + 1/q = 1$ | $\|fg\|_1 \leq \|f\|_p \|g\|_q$ | | Minkowski’s Inequality | $f, g \in L^p$ | $\|f + g\|_p \leq \|f\|_p + \|g\|_p$ | | Fubini | $f \in L^1(\mu \times \nu)$ | Iterated integrals equal double integral | | Radon-Nikodym | $\nu \ll \mu$, $\sigma$-finite | $d\nu/d\mu$ exists and is unique a.e. |

Statement. If $\{f_n\}$ is a sequence of measurable functions such that $0 \leq f_1 \leq f_2 \leq \cdots$ and $f_n \to f$ pointwise, then $\int f\,d\mu = \lim_{n\to\infty} \int f_n\,d\mu$.

Intuition. The monotone convergence theorem (MCT) allows the interchange of limit and integral when the functions are non-negative and increase monotonically. It is the primary tool for proving convergence of integrals when the integrands are positive.

Application. To compute $\int_0^\infty x^{p-1} e^{-x}\,dx = \Gamma(p)$, one expresses the integrand as the increasing limit of functions $x^{p-1} e^{-x} \mathbf{1}_{[0,n]}(x)$ and applies MCT.

Statement. If $\{f_n\}$ is a sequence of non-negative measurable functions, then $\int \liminf_{n\to\infty} f_n\,d\mu \leq \liminf_{n\to\infty} \int f_n\,d\mu$.

Intuition. Fatou’s lemma gives a one-sided inequality that holds without any convergence assumptions. It is often used as a stepping stone to prove the dominated convergence theorem.

Common Pitfall. The inequality can be strict. For example, $f_n = \mathbf{1}_{[n,n+1]}$ on $\mathbb{R}$ with Lebesgue measure: $\liminf f_n = 0$ so the left side is $0$, but $\int f_n = 1$ for each $n$, so the right limit inferior is $1$.

Statement. Suppose $f_n \to f$ pointwise a.e. and $|f_n| \leq g$ for all $n$ where $g \in L^1$. Then $\int f_n \to \int f$.

Intuition. The dominated convergence theorem (DCT) is the workhorse of measure theory. It allows interchange of limit and integral provided the functions are dominated by an integrable function. The dominating function must be in $L^1$, not merely bounded.

Application. Continuity of the Laplace transform: for $F(t) = \int_0^\infty e^{-tx} f(x)\,dx$, if $f \in L^1$, then $F$ is continuous on $[0,\infty)$ by DCT.

Statement. For measurable functions $f$ and $g$, if $f \in L^p$ and $g \in L^q$ where $1/p + 1/q = 1$, then $fg \in L^1$ and $\int |fg| \leq \|f\|_p \|g\|_q$.

Intuition. H”older’s inequality generalises the Cauchy-Schwarz inequality (the special case $p = q = 2$). It is fundamental for establishing duality between $L^p$ and $L^q$ spaces.

Proof Sketch. By Young’s inequality $ab \leq a^p/p + b^q/q$ for $a, b \geq 0$. Apply this to $a = |f|/\|f\|_p$, $b = |g|/\|g\|_q$ and integrate.

Statement. For $f, g \in L^p$ with $1 \leq p \leq \infty$, we have $\|f + g\|_p \leq \|f\|_p + \|g\|_p$.

Intuition. Minkowski’s inequality is the triangle inequality for the $L^p$ norm. It establishes that $L^p$ is a normed vector space. For $p < 1$, the reverse inequality holds.

Proof Sketch. Write $|f+g|^p \leq |f+g|^{p-1}|f| + |f+g|^{p-1}|g|$, apply H”older with exponents $p$ and $p/(p-1)$, then divide.

Statement. Let $f \in L^1(\mu \times \nu)$ on a product measure space. Then: $\int_{X \times Y} f\,d(\mu \times \nu) = \int_X \left(\int_Y f(x,y)\,d\nu(y)\right) d\mu(x) = \int_Y \left(\int_X f(x,y)\,d\mu(x)\right) d\nu(y)$.

Intuition. Fubini’s theorem justifies interchanging the order of integration. The key condition is absolute integrability with respect to the product measure.

Common Pitfall. Without $L^1$ condition, the iterated integrals may differ. The classic counterexample is $f(x,y) = (x^2 - y^2)/(x^2 + y^2)^2$ on $[0,1]^2$, where iterated integrals are $\pm \pi/4$.

Statement. If $\nu \ll \mu$ (absolutely continuous) and both are $\sigma$-finite, then there exists a measurable function $f = d\nu/d\mu$ such that $\nu(A) = \int_A f\,d\mu$ for all measurable $A$. The function $f$ is unique almost everywhere.

Intuition. The Radon-Nikodym derivative generalises the idea of a density function from probability theory. For example, if $\nu$ has density $f$ with respect to Lebesgue measure, then $d\nu = f\,dx$.

Application. Conditional expectation in probability theory: $E[X|\mathcal{F}]$ is defined as the Radon-Nikodym derivative $d\nu/dP$ where $\nu(A) = \int_A X\,dP$ for $A \in \mathcal{F}$.

The convergence theorems (MCT, Fatou, DCT) form an interdependent chain: MCT implies Fatou, and Fatou plus a dominating function implies DCT. The inequalities (H”older, Minkowski) define the geometry of $L^p$ spaces. Fubini connects product integration to iterated integration. Radon-Nikodym bridges absolute continuity and densities.

1. Compute $\lim_{n\to\infty} \int_0^\infty \frac{\sin(x/n)}{x(1+x^2)}\,dx$ using DCT.
2. Show that if $f_n \to f$ in $L^p$ then $\int |f_n|^p \to \int |f|^p$. Give a counterexample where $f_n \to f$ a.e. but $\int f_n \not\to \int f$ without a dominating function.
3. Let $f \in L^1(\mathbb{R})$ and define $F(x) = \int_{\mathbb{R}} f(t) e^{-ixt}\,dt$. Show $F$ is continuous using DCT.
4. Use H”older’s inequality to prove that if $\mu(X) < \infty$, then $L^p(X) \subseteq L^q(X)$ for $1 \leq q \leq p \leq \infty$.
5. Find a measurable function $f$ on $[0,1]^2$ such that $\int_0^1 \int_0^1 f\,dx\,dy \neq \int_0^1 \int_0^1 f\,dy\,dx$, and identify why Fubini’s theorem does not apply.

Measure theory provides rigorous foundations for integration. The convergence theorems (MCT, Fatou, DCT) govern when limits and integrals can be interchanged. H”older and Minkowski inequalities establish $L^p$ space structure. Fubini’s theorem handles product measures, and the Radon-Nikodym theorem connects measures via densities.