Subset of the preimage of a semicontinuous real function is Borel
A real function $f$ on the line is upper semi-continuous at $x$, if for each $\epsilon > 0$, there exists $\delta > 0$ such that $|x-y|<\delta$ implies that $f(y) < f(x) + \epsilon$. Check that if $f$ is everywhere upper semi-continuous, then it is measurable.
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