The Unapologetic Mathematician

Mathematics for the interested outsider

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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