# Lebesgue Integral

For this entire section we fix a measure space $(X,A,μ)$.

## Integration of Nonnegative Step Functions

Let $f:X→R$ be a nonnegative step function with representation $f=∑_{i=1}α_{i}χ_{A_{i}}$, where $α_{1},…,α_{n}≥0$ and $A_{1},…,A_{n}∈A$. We define the

$∫_{X}fdμ=i=1∑n α_{i}μ(A_{i})∈[0,∞].$integral of $f$ on $X$ with respect to $μ$by

TODO: This does not depend on the representation of $f$.

## Integration of Nonnegative Measurable Functions

Theorem (Approximation by Step Functions)

Every nonnegative measurable function $f:X→R$ is the pointwise limit of an increasing sequence $(s_{n})$ of nonnegative step functions $s_{n}:X→R$.

Let $f:X→R$ be a nonnegative measurable function and let $(s_{n})$ be a sequence of nonnegative step functions with $s_{n}↑f$. We define the

$∫_{X}fdμ=n→∞lim ∫_{X}s_{n}dμ∈[0,∞].$integral of $f$ on $X$ with respect to $μ$by

## Integrable Functions

Recall that the positive and (flipped) negative parts of a function $f:X→R$ are defined by

$f_{+}=max(f,0)f_{−}=max(−f,0),$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 $f:X→R$ is said to be

$∫_{X}f_{+}dμ,∫_{X}f_{−}dμ(∗)$integrable on $X$ with respect to $μ$if the integralsare both finite. In this case the

$∫_{X}fdμ=∫_{X}f_{+}dμ−∫_{X}f_{−}dμ∈R.$(Lebesgue) integral of $f$ on $X$ with respect to $μ$is defined as

Sometimes it is convenient to have a slightly more general notion of integrability:

Definition (Quasi-Integrable Function)

A measurable function $f:X→R$ is said to be

$∫_{X}fdμ=∫_{X}f_{+}dμ−∫_{X}f_{−}dμ∈R.$quasi-integrable on $X$ with respect to $μ$if at least one of the integrals $(∗)$ is finite. In this case theintegral of $f$ on $X$ with respect to $μ$is defined as

A measurable function $f:X→C$ is said to be

$∫_{X}fdμ=∫_{X}Refdμ+i∫_{X}Imfdμ∈C.$integrable on $X$ with respect to $μ$if $Ref$ and $Imf$ are integrable on $X$ with respect to $μ$. In this case theintegral of $f$ on $X$ with respect to $μ$is defined as

## Integration on Measurable Subsets

For any measurable subset $A⊂X$ we define the

$∫_{A}fdμ=∫_{X}χ_{A}fdμ.$integral on $A$of a (quasi-)integrable function $f:X→R$ (or $C$) by