# Prove that Borel sigma field on R(d) is the smallest sigma-field that makes all continuous functions f:R(d)->R measurable.

Prove that Borel sigma-field on $\mathbb{R}^d$ is the smallest sigma-field that makes all continuous functions $f:\mathbb{R}^d \to \mathbb{R}$ measurable.

How do I go about proving this? Thanks!

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It is only a matter of definitions. Recall that a continuous function is one for which the preimage of an open set is an open set and that the Borel sigma-field is the smallest sigma-field containing the open sets. –  Giuseppe Negro Sep 1 '12 at 8:09
To show that the smallest $\sigma$-algebra making all continuous functions measurable is at least as large as the Borel $\sigma$-algebra, you can use that $\mathbb{R}^n$ is perfectly normal. –  Michael Greinecker Sep 1 '12 at 10:34
Let $F\subseteq\mathbb{R}^n$ be a nonempty closed set. We let $\delta_F:\mathbb{R}^n\to\mathbb{R}$ be the distance from $F$, that is, $$\delta_F(x)=\inf\big\{\|y-x\|:y\in F\big\}$$ for all $x\in\mathbb{R}^n$. You can show that $\delta_F(x)=0$ if and only if $x\in F$, so $F=\delta_F^{-1}\big(\{0\}\big)$, and $\{0\}$ is a Borel subset of $\mathbb{R}$. So all closed sets are in the smallest $\sigma$-field that makes all continuous functions measurable. The rest should be manageable.