The Unapologetic Mathematician

Measurable (Extended) Real-Valued Functions

For a while, we’ll mostly be interested in real-valued functions with Lebesgue measure on the real line, and ultimately in using measure to give us a new and more general version of integration. When we couple this with our slightly weakened definition of a measurable space, this necessitates a slight tweak to our definition of a measurable function.

Given a measurable space $(X,\mathcal{S})$ and a function $f:X\to\mathbb{R}$, we define the set $N(f)$ as the set of points $x\in X$ such that $f(x)\neq0$. We will say that the real-valued function $f$ is measurable if $N(f)\cap f^{-1}(M)$ is a measurable subset of $X$ for every Borel set $M\in\mathcal{B}$ of the real line. We have to treat ${0}$ specially because when we deal with integration, ${0}$ is special — it’s the additive identity of the real numbers.

The entire real line $\mathbb{R}$ is a Borel set, and $f^{-1}(\mathbb{R})=X$. Thus we find that $N(f)$ must be a measurable subset of $X$. If $E$ is another measurable subset of $X$, then we observe

$\displaystyle E\cap f^{-1}(M)=((E\cap N(f))\cap f^{-1}(M))\cup((E\setminus N(f))\cap f^{-1}(M))$

The second term on the right is either empty or is equal to $E\setminus N(f)$. And so it’s clear that $E\cap f^{-1}(M)$ is measurable. We say that the function $f$ is “measurable on $E$” if $E\cap f^{-1}(M)$ is measurable for every Borel set $M$, and so we have shown that a measurable function is measurable on every measurable set.

In particular, if $X$ is itself measurable (as it often is), then a real-valued function is measurable if and only if $f^{-1}(M)$ is measurable for every Borel set $M\in\mathcal{B}$. And so in this (common) case, we get back our original definition of a measurable function $f:(X,\mathcal{S})\to(\mathbb{R},\mathcal{B})$.

The concept of measurability depends on the $\sigma$-ring $\mathcal{S}$, and we sometimes have more than one $\sigma$-ring floating around. In such a case, we say that a function is measurable with respect to $\mathcal{S}$. In particular, we will often be interested in the case $X=\mathbb{R}$, equipped with either the $\sigma$-algebra of Borel sets $\mathcal{B}$ or that of Lebesgue measurable sets $\overline{\mathcal{B}}$. A measurable function $f:(\mathbb{R},\mathcal{B})\to(\mathbb{R},\mathcal{B})$ will be called “Borel measurable”, while a measurable function $f:(\mathbb{R},\overline{\mathcal{B}})\to(\mathbb{R},\mathcal{B})$ will be called “Lebesgue measurable”.

On the other hand, we should again emphasize that the definition of measurability does not depend on any particular measure $\mu$.

We will also sometimes want to talk about measurable functions taking value in the extended reals. We take the convention that the one-point sets $\{\infty\}$ and $\{-\infty\}$ are Borel sets; we add the requirement that a real-valued function also have $f^{-1}(\{\infty\})$ and $f^{-1}(\{-\infty\})$ both be measurable to the condition for $f$ to be measurable. However, for this extended concept of Borel sets, we can no longer generate the class of Borel sets by semiclosed intervals.

April 30, 2010 - Posted by | Analysis, Measure Theory

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