Lebesgue Integration - Key Properties

The Lebesgue integral possesses several fundamental properties that make it the cornerstone of modern analysis. These properties demonstrate its superiority over the Riemann integral for many applications.

TheoremLinearity of the Integral

Let $f$ and $g$ be integrable functions on $(X, \mathcal{F}, \mu)$ and let $\alpha, \beta \in \mathbb{R}$ . Then $\alpha f + \beta g$ is integrable and: $\int_X (\alpha f + \beta g) \, d\mu = \alpha \int_X f \, d\mu + \beta \int_X g \, d\mu$

This shows that integration is a linear functional on the space of integrable functions.

ExampleMonotonicity and Triangle Inequality

Key inequalities for the Lebesgue integral include:

Monotonicity: If $f \leq g$ almost everywhere and both are integrable, then: $\int_X f \, d\mu \leq \int_X g \, d\mu$
Triangle inequality: If $f$ is integrable, then: $\left|\int_X f \, d\mu\right| \leq \int_X |f| \, d\mu$
Absolute integrability: $f$ is integrable if and only if $|f|$ is integrable.

These properties mirror familiar properties of sums and series.

Remark

Important operational properties include:

Domain additivity: If $A$ and $B$ are disjoint measurable sets, then: $\int_{A \cup B} f \, d\mu = \int_A f \, d\mu + \int_B f \, d\mu$

Measure zero sets: If $\mu(E) = 0$ and $f$ is measurable, then: $\int_E f \, d\mu = 0$

Almost everywhere equality: If $f = g$ a.e. and $f$ is integrable, then $g$ is integrable and: $\int_X f \, d\mu = \int_X g \, d\mu$

This last property is crucial: the integral depends only on the equivalence class of $f$ modulo equality almost everywhere.

DefinitionL1 Space

The space of integrable functions is denoted $L^1(X, \mathcal{F}, \mu)$ or simply $L^1(\mu)$ . More precisely, $L^1$ consists of equivalence classes of functions that are equal almost everywhere, with the norm: $\|f\|_1 = \int_X |f| \, d\mu$

This makes $L^1$ a Banach space (a complete normed vector space), which is fundamental for functional analysis.

The completeness of $L^1$ means that every Cauchy sequence of integrable functions converges to an integrable function, making $L^1$ a natural setting for solving equations and optimization problems.