Let $f(x)$ be a real measurable function defined on $[a,b]$. Let $n(y)$ be the number of solutions of the equation $f(x) =y$. Prove that $n(y)$ is a measurable function on $\mathbb{R}$. Considering f(x)=a is measureable and discuss the counting measure of it?
A Problem for measure function $n(y)$,the number of solution
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real-analysis
measure-theory
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0@GregMartin I have not consider that.But I think it should make no difference. – 2012-12-12