[Math] Proof verification (to show that every continuous function is measurable)


The problem is:

Show that every continuous function $f: \mathbb{R}^d \to [0,+\infty]$ is measurable.

I'm trying to use the following open set formulation of measurable function:

For all open set $U \subset [0,+\infty)$, $f^{-1}(U)$ is measurable.

Now if $f$ is continuous, then for all open set $U \subset [0,+\infty)$, $f^{-1}(U)$ is open. We know that all open sets are measurable. Hence the open set formulation is satisfied and we see that $f$ is measurable.

The proof seems too straightforward and I'm a bit suspicious about holes in it. Any help, confirmation or correction, would be greatly appreciated.

Best Answer

This proof is fine.

The proof seems too straightforward, because what you are trying to prove is elementary, so it does not require sophisticated argument. Since measurable functions are much more general then continuous functions, it is normal that it is easy to prove that continuous functions are measurable.

Great job!

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