Analysis Problem of the Day 98

Today’s problem appeared as Problem 4 on the Texas A&M August 2022 Analysis Qual:

Problem 4. Either give an example of a $\sigma$ -finite measure space that can be written as a disjoint union of uncountably many measurable sets of positive measure, or prove that such a space cannot exist.

Solution: We claim that such a space cannot exist. This follows from the following intuition: a $\sigma$ -finite measure space is at most a countable union of finite measure spaces, and a finite measure spaces is at most a countable union of disjoint sets of positive measure (since uncountable sums of positive numbers have infinite value). Formally, we proceed as follows. Let $(X,\mathcal{M},\mu)$ be the given measure space, and write $X = \bigcup_{i=1}^\infty X_i, \mu(X_i)<\infty.$ Suppose $X$ can be written as a disjoint union of uncountably many measurable sets $\{A_\lambda\}_{\lambda \in \mathbb{R}}.$ We claim that at most countably many of these sets have positive measure. Indeed, note that for some fixed $X_i,$ $\mu(A_\lambda \cap X_i)>0$ for at most countably many $\lambda.$ Otherwise, there exists an $n$ such that $\mu(A_\lambda \cap X_i)>\frac{1}{n}$ holds for countably infinitely many $\lambda,$ implying by disjointness of the $A_\lambda$ that $\mu(X_i) =\infty.$ Since $\mu(A_\lambda)>0$ if and only if $\mu(A_\lambda \cap X_i)>0$ for some $i,$ it follows that the set of $\lambda \in \mathbb{R}$ such that $\mu(A_\lambda)>0$ is at most a countable union of countable sets (those such that $\mu(A_\lambda \cap X_i) >0$ for each $i=1,2,...$ ) and is therefore at most countable, proving the desired claim.

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