Lebesgue Measurable Functions

Simple Functions and Measurable Functions

Characteristic Function

A characteristic function of a set $E$ is a function defined by ${\chi }_E(x)=\{\begin{array}{l}1,\text{ if }x\in E\\ 0,\text{if }x\notin E\end{array}$

Simple Function, Step Function

A simple function is a function that can be expressed as a finite linear combination of characteristic functions of measurable sets with finite measure: $f={\sum }_{k=1}^N{a}_k{\chi }_{E_k}$where $a_k$ are constants. In particular, if each $E_k$ is a rectangle, then $f$ is called a step function.

Lebesgue Measurable Function

A function $f:E\to [-\infty ,\infty ]$ defined on a measurable subset $E$ of ${\mathbb{R}}^d$ is Lebesgue measurable, if the set $f^{-1}(B)$ is Lebesgue measurable for all Borel set $B\subseteq \mathbb{R}$. To simplify our notation, we shall often denote the set $\{x\in E:f(x)<a\}$ simply by $\{f<a\}$ whenever no confusion is possible.

Proposition

The followings are equivalent:

1. $f$ is Lebesgue measurable.
2. $\{f\le a\}$ or $\{f\ge a\}$ or $\{f<a\}$ or $\{f>a\}$ is measurable for all $a\in \mathbb{R}$.

Proposition

The followings are equivalent for a finite-valued function $f$:

1. $f$ is measurable.
2. $\{a\le f<b\}$ is measurable for all $a,b\in \mathbb{R}$.
3. $f^{-1}(O)$ is measurable for all open sets $O\subset \mathbb{R}$.
4. $f^{-1}(C)$ is measurable for all closed sets $C\subset \mathbb{R}$.

Proposition

If $f$ is continuous on ${\mathbb{R}}^d$, then $f$ is measurable. If $f$ is measurable and finite-valued, and $\Phi$ is continuous, then $\Phi \circ f$ is measurable.

Proposition

Suppose $\{f_n{\}}_{n=1}^{\infty }$ is a sequence of measurable functions. Then $x\mapsto {\sup }_n{f}_n(x),x\mapsto {\inf }_n{f}_n(x),{\mathrm{limsup}}_{n\to \infty }{f}_n(x),{\mathrm{liminf}}_{n\to \infty }{f}_n(x)$are measurable. In particular, if ${\lim }_{n\to \infty }{f}_n$ exists a.e., it is measurable.

Proof Note that $\{{\sup }_n{f}_n>a\}={\bigcup }_n\{f_n>a\}$, and $\{{\mathrm{limsup}}_n{f}_n>a\}={\bigcap }_N{\bigcup }_{n\ge N}\{f_n>a\}$, and other cases are similar. $\square$

Proposition

If $f$ and $g$ are measurable, then

1. The integer powers $f^k$, $k\ge 1$ are measurable.
2. $f+g$ and $fg$ are measurable if both $f$ and $g$ are finite-valued.

Almost Everywhere

We say that a property holds almost everywhere (a.e.) if the set of points where the property fails is a null set.

e.g. We say functions $f=g$ a.e. $x\in E$ if $\{x\in E:f(x)\ne g(x)\}$ has (outer) measure zero.

Proposition

Suppose $f$ is measurable, and $f(x)=g(x)$ a.e. $x\in E$. Then $g$ is measurable.

Approximation by Simple Functions

Theorem

Suppose $f$ is a non-negative measurable function on ${\mathbb{R}}^d$. Then there exists an increasing sequence of non-negative simple functions $\{{\phi }_k{\}}_{k=1}^{\infty }$ that converges pointwise to $f$, namely,
${\phi }_k(x)\le {\phi }_{k+1}(x)\text{and}{\lim }_{k\to \infty }{\phi }_k(x)=f(x),\text{for all }x.$

Corollary

Suppose $f$ is measurable on ${\mathbb{R}}^d$. Then there exists a sequence of simple functions $\{{\phi }_k{\}}_{k=1}^{\infty }$ that satisfies
$|{\phi }_k(x)|\le |{\phi }_{k+1}(x)|\text{and}{\lim }_{k\to \infty }{\phi }_k(x)=f(x),\text{ for all }x.$ In particular, we have $|{\phi }_k(x)|\le |f(x)|$ for all $x$ and $k$.

We may now go one step further, and approximate by step functions. Here, in general, the convergence may hold only almost everywhere.

Theorem

Suppose $f$ is measurable on ${\mathbb{R}}^d$. Then there exists a sequence of step functions $\{{\psi }_k{\}}_{k=1}^{\infty }$ that converges pointwise to $f(x)$ for almost every $x$.