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How do I show that for $f,g\in C[a,b]$ the set $A=\{(x,y)\in \mathbb{R}^2:a\leq x\leq b, f(x)\leq y\leq g(x)\}$ is Lebesgue-measurable?

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Since $f $ and $g $ are continuous, $A $ is a closed subset of $\mathbb R^2,$ which gives the assertion.

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