From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 14 Feb 2024 07:24:38 +0100
Subject: Update

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+---
+title: Lebesgue Integral
+parent: Measure and Integration
+nav_order: 2
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+For this entire section we fix a measure space $(X,\mathcal{A},\mu)$.
+
+## Integration of Nonnegative Step Functions
+
+{% definition %}
+Let $f : X \to \RR$ be a nonnegative step function
+with representation $f = \sum_{i=1}^n \alpha_i \chi_{A_i}$,
+where $\alpha_1, \ldots, \alpha_n \ge 0$ and
+$A_1, \ldots, A_n \in \mathcal{A}$.
+We define the *integral of $f$ on $X$ with respect to $\mu$* by
+
+$$
+\int_X f \, d\mu = \sum_{i=1}^n \alpha_i \, \mu(A_i) \in [0,\infty].
+$$
+{% enddefinition %}
+
+TODO: This does not depend on the representation of $f$.
+
+## Integration of Nonnegative Measurable Functions
+
+{% theorem Approximation by Step Functions %}
+Every nonnegative measurable function $f : X \to \overline{\RR}$
+is the pointwise limit of an increasing sequence $(s_n)$ of
+nonnegative step functions $s_n : X \to \RR$.
+{% endtheorem %}
+
+{% definition %}
+Let $f : X \to \overline{\RR}$ be a nonnegative measurable function
+and let $(s_n)$ be a sequence of nonnegative step functions
+with $s_n \uparrow f$.
+We define the *integral of $f$ on $X$ with respect to $\mu$* by
+
+$$
+\int_X f \, d\mu = \lim_{n \to \infty} \int_X s_n \, d\mu \in [0,\infty].
+$$
+{% enddefinition %}
+
+## Integrable Functions
+
+Recall that the positive and (flipped) negative parts
+of a function $f : X \to \overline{R}$ are defined by
+
+$$
+f^+ = \max(f,0) \qquad
+f^- = \max(-f,0),
+$$
+
+and that $f$ is measurable if and only if both $f^+$ and $f^-$ are measurable.
+We have $f = f^+\! - f^-$.
+
+{% definition Integrable Function, Lebesgue Integral %}
+A measurable function $f : X \to \overline{\RR}$ is said to be
+*integrable on $X$ with respect to $\mu$* if the integrals
+
+$$
+\int_X f^+ \, d\mu, \qquad \int_X f^- \, d\mu
+\tag{$*$}
+$$
+
+are both finite.
+In this case the *(Lebesgue) integral of $f$ on $X$ with respect to $\mu$* is defined as
+
+$$
+\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \RR.
+$$
+{% enddefinition %}
+
+Sometimes it is convenient to have a slightly more general notion of integrability:
+
+{% definition Quasi-Integrable Function %}
+A measurable function $f : X \to \overline{\RR}$ is said to be
+*quasi-integrable on $X$ with respect to $\mu$* if at least one of the integrals
+$(*)$ is finite.
+In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as
+
+$$
+\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \overline{\RR}.
+$$
+{% enddefinition %}
+
+{% definition %}
+A measurable function $f : X \to \CC$ is said to be
+*integrable on $X$ with respect to $\mu$* if
+$\Re f$ and $\Im f$ are integrable on $X$ with respect to $\mu$.
+In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as
+
+$$
+\int_X f \, d\mu = \int_X \Re f \, d\mu + i \int_X \Im f \, d\mu \in \CC.
+$$
+{% enddefinition %}
+
+## Integration on Measurable Subsets
+
+{% definition %}
+For any measurable subset $A \subset X$ we define
+the *integral on $A$* of a (quasi-)integrable function $f : X \to \overline{\RR}$ (or $\CC$) by
+
+$$
+\int_A f \, d\mu = 
+\int_X \chi_A f \, d\mu.
+$$
+{% enddefinition %}
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