3  Integration

Published

December 9, 2025

3.1 Properties

3.1.1 Definition

Let (E, \mathcal E, \mu) be a measure space. For measurable non-negative f

\mu[f] = sup_{g \text{ simple }, g \leq f} \int g d \mu = \lim_{n \to \infty} \int d_n \circ f d\mu

where the integral of a simple function g = \sum_{n=1}^N \mathbb a_i 1_{A_i} is defined as \mu [g] = \sum_{n=1}^N a_i \mu(A_i)

For measurable functions of arbitrary sign, f = f^+ + f^-

3.1.2 Linearity

\mu[a f + bg] = a \mu[f] + b\mu[g]

Proof \begin{align*} \mu[a f + bg] &= \mu[\lim a f_n + bg_n]\\ &= \lim \mu [a f_n + b g_n] \qquad \text{MCT}\\ &= \lim a \mu [f_n] + \lim b \mu[g_n] \qquad \text{linearity of simple fn integrals}\\ &= a \mu[f] + b \mu[g] \qquad \text{MCT}\\ \end{align*}

3.1.3 Monotonicity

f \geq g \implies \mu[f] \geq \mu[g]

3.1.4 Insensitivity

  1. For f \in \mathcal E_+ ,\mu[f] = 0 \iff f = 0 \mu-a.e.

  2. For all measurable f, \mu(A) = 0 i.e. \mathbb 1_A =0 \quad\mu- a.e. \implies \int_A f(x) d\mu = \mu[\mathbb 1_A f] = 0

  3. If f = g \quad \mu-a.e. then \mu[f] = \mu[g].

  4. If \mu[f] < \infty then f < \infty \quad \mu-a.e.

  5. \mu[|f-g|] = 0 or \mu[(f-g)^2] = 0 \implies f = g \mu-a.e.

  6. If for all measurable sets A, \mu[f \mathbb 1_A] = \mu[g \mathbb 1_A] then f = g \mu-a.e.

  7. If for all positive measurable functions p, \mu[fp] = \mu[gp] then f = g \mu-a.e.

3.1.5 Characterisation theorem

Let (E, \mathcal E) be a measurable space, and define a functional L: \mathcal E_+ \to \mathbb R. Then there exists a measure \mu on (E, \mathcal E) such that for all f \in \mathcal E_+, L[f] = \mu[f] iff it satisfies:

  1. Linearity: For all a, b \in \mathbb R_+, f, g \in \mathcal E_+, L[af + bg] = a L[f] + b L[g].
  2. Monotone convergence: (f_n) \subset \mathcal E_+ and f_n \nearrow f \implies L[f_n] \nearrow L[f]
Proof

3.1.5.1 Proving equivalence of a functional and integral

If I want to prove that a functional L is equal to a Lebesgue integral for some specific measure \mu i.e. L[f] = \mu[f] \quad \forall f \in \mathcal E_+:

  1. Prove L satisfies linearity and monotone convergence. Thus there exists a unique measure \quad \tilde \mu such that \tilde \mu [f] = L[f] for all f\in \mathcal E_+
  2. For all B \in \mathcal E (or a p-system generator), show \tilde \mu(B) = L[\mathbb 1_B] = \mu(B).

These imply \tilde \mu = \mu and so \tilde \mu [f] = L[f] = \mu[f]

3.2 Limits and integrals

3.2.1 Monotone convergence theorem

Let (f_n) \subset \mathcal E_+ be a monotone increasing sequence of positive measurable functions. Then, \mu [\lim f_n] = \lim \mu[f_n]

Equivalently, if (g_n) is a sequence of positive measurable functions:

\mu[\sum_n g_n] = \sum_n \mu[g_n]

3.2.2 Fatou’s lemma

For general sequences f_n \subset \mathcal E of measurable functions: \mu [\lim f_n] \leq \lim \inf \mu [f_n]

3.2.3 Fatou-Lebesgue theorem

Let (f_n) be a sequence of measurable functions. If there exists an integrable g that dominates the sequence in absolute value i.e. |f_n| \leq g for all n, then all f_n are integrable and: \mu [\lim f_n] \leq \lim \inf \mu [f_n] \leq \lim \sup \mu [f_n] \leq \mu [\lim\sup f_n]

3.2.4 Dominated convergence theorem

Suppose (f_n) is dominated by some integrable g. If the limit exists, then:

\mu [\lim f_n] = \lim \mu[f_n]

3.2.5 Leibniz integral rule

For measure space (E, \mathcal E, \mu) consider f: E \times \mathbb R \to \mathbb R, g(\theta) = \int f(x, \theta) \mu(dx). Fix \theta_0 \in \mathbb R and an open interval U with \theta_0 \in U.

If there exists an integrable h : E \to \mathbb R such that for almost all x and all \theta \in U, \frac{\partial f}{\partial \theta} exists and \left|\frac{\partial f}{\partial \theta}(x, \theta)\right| \leq h(\theta)

Then \frac{\partial g}{\partial \theta}(\theta_0) exists and is: \frac{\partial g}{\partial \theta}(\theta_0) = \int \frac{\partial f}{\partial \theta}(x, \theta_0) \mu(dx)

3.3 Fubini’s theorem

Let \mu, \nu be \Sigma- finite measures on (E, \mathcal E), (F, \mathcal F) respectively. Then, there exists a unique \Sigma- finite measure \pi on the product space (E \times F, \mathcal E \otimes \mathcal F) such that for every positive function f \in (\mathcal E \otimes \mathcal F)_+ measurable wrt the product sigma algebra:

\pi [f] = \int \mu(dx) \int \nu(dy) f(x, y) = \int \nu(dy) \int \mu(dx) f(x,y)

For A \in \mathcal E, B \in \mathcal F, \pi( A \times B) = \mu(A) \nu(B). If \mu, \nu are \sigma- finite, then \pi is uniquely determined by this p-system generator.