Subset of the preimage of a semicontinuous real function is Borel

real-analysis measure-theory borel-sets semicontinuous-functions

Observe that $\{x\,:\,f(x) \geq a\} = \bigcap_{n=1}^{\infty} \left\{x \,:\,f(x) \gt a-\frac{1}{n}\right\}$, and the latter is a countable intersection of open sets by definition of lower semicontinuity.

Added later: See also this related thread for an argument using a different definition of (upper) semicontinuity.

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