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Let $f : X → \overline R$ be measurable on a $σ$-finite measure space $(X, A, µ)$. Show that then the set ${x ∈ R : µ( f^{−1}(\{x\}) ) > 0}$ is countable.

Firstly, I noted that "$σ$-finite measure space $(X, A, µ)$" means X = $\bigcup\limits_{i=1}^{\infty} A_{i}$ (i.e the countable union of) for some sequence of measurable sets $\{A_{i}\}\in A$ where µ($A_{i}) < \infty,\forall i\geq 1$.

Secondly, $f$ is measurable. Therefore $f^{-1}((a,\infty]) = \{x\in X:f(x)>a\}\in A ,\forall a\in R$.

I can see that there will be the need to relate the countablity mentioned above to the set in question via unions and intersections of sets, but I cannot see how to proceed.

Any suggestions would be great.

TheWaveMan
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  • Look here for inspiration how to solve it: http://math.stackexchange.com/questions/13781/transfinite-series-uncountable-sums – Michael Greinecker Apr 04 '17 at 13:53
  • Another question posed very recently seems related to yours: http://math.stackexchange.com/questions/2217454/an-uncountable-sum-that-equals-1-but-could-not-be-written-as-a-countable-subsu – mlc Apr 04 '17 at 14:02

1 Answers1

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I would begin by showing that, for fixed $n \in \mathbb N$ and for fixed $i \in \mathbb N$, the set $$ E_{n,i} = \{ x \in \mathbb R : \mu(f^{-1}(\{ x \}) \cap A_i) > 1/n \}$$ is a finite set.

[The fact that $\mu(A_i) < \infty$ is key here.]

Then try to show that if $\mu(f^{-1}(\{ x \}) > 0$, then we must have $x \in E_{n,i}$ for some $n$ and some $i$.

Is this enough of a hint?

Kenny Wong
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