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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?

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Are you allowing the possibility that $n(y) = +\infty$? –  Greg Martin Dec 12 '12 at 18:39
@GregMartin I have not consider that.But I think it should make no difference. –  Jebei Dec 12 '12 at 18:41

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