Lebesgue Integral

For this entire section we fix a measure space $(X,\mathcal{A},\mu )$.

Integration of Nonnegative Step Functions

Let $f:X\to \mathbb{R}$ be a nonnegative step function with representation $f={\sum }_{i=1}^n{\alpha }_i{\chi }_{A_i}$, where ${\alpha }_1,\dots ,{\alpha }_n\ge 0$ and $A_1,\dots ,A_n\in \mathcal{A}$. We define the integral of $f$ on $X$ with respect to $\mu$ by

$${\int }_{X}fd\mu =\sum _{i=1}^n{\alpha }_i\mu (A_i)\in [0,\infty ].$$

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

Integration of Nonnegative Measurable Functions

Theorem (Approximation by Step Functions)

Every nonnegative measurable function $f:X\to \overline{\mathbb{R}}$ is the pointwise limit of an increasing sequence $(s_n)$ of nonnegative step functions $s_n:X\to \mathbb{R}$.

Let $f:X\to \overline{\mathbb{R}}$ 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

$${\int }_{X}fd\mu =\underset{n\to \infty }{\lim }{\int }_X{s}_{n}d\mu \in [0,\infty ].$$

Integrable Functions

Recall that the positive and (flipped) negative parts of a function $f:X\to \overline{R}$ are defined by

$$f^{+}=\max (f,0)f^{-}=\max (-f,0),$$

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 $f:X\to \overline{\mathbb{R}}$ is said to be integrable on $X$ with respect to $\mu$ if the integrals

$${\int }_X{f}^{+}d\mu ,{\int }_X{f}^{-}d\mu \text{\hspace{1em}(∗)}$$

are both finite. In this case the (Lebesgue) integral of $f$ on $X$ with respect to $\mu$ is defined as

$${\int }_{X}fd\mu ={\int }_X{f}^{+}d\mu -{\int }_X{f}^{-}d\mu \in \mathbb{R}.$$

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

Definition (Quasi-Integrable Function)

A measurable function $f:X\to \overline{\mathbb{R}}$ is said to be quasi-integrable on $X$ with respect to $\mu$ if at least one of the integrals $(\ast )$ is finite. In this case the integral of $f$ on $X$ with respect to $\mu$ is defined as

$${\int }_{X}fd\mu ={\int }_X{f}^{+}d\mu -{\int }_X{f}^{-}d\mu \in \overline{\mathbb{R}}.$$

A measurable function $f:X\to \mathbb{C}$ is said to be integrable on $X$ with respect to $\mu$ if $\mathrm{Re}f$ and $\mathrm{Im}f$ are integrable on $X$ with respect to $\mu$. In this case the integral of $f$ on $X$ with respect to $\mu$ is defined as

$${\int }_{X}fd\mu ={\int }_X\mathrm{Re}fd\mu +i{\int }_X\mathrm{Im}fd\mu \in \mathbb{C}.$$

Integration on Measurable Subsets

For any measurable subset $A\subset X$ we define the integral on $A$ of a (quasi-)integrable function $f:X\to \overline{\mathbb{R}}$ (or $\mathbb{C}$) by

$${\int }_{A}fd\mu ={\int }_X{\chi }_{A}fd\mu .$$