Show that every decreasing function $f$: $\mathbb{R} \rightarrow \mathbb{R}$ is $\mathcal{B}(\mathbb{R})$ measurable

Question

I have the following question from an elementary course on Measure Theory:

Show that every decreasing function $f$: $\mathbb{R} \rightarrow \mathbb{R}$ is $\mathcal{B}(\mathbb{R})\space/ \space \mathcal{B}(\mathbb{R})$ measurable (where $\mathcal{B}(\mathbb{R})$ is the Borel Sigma Algebra on $\mathbb{R}$)

Any hints to prove this result would be appreciated :)

You could have a slightly less general title. Measure theory is a fairly broad subject. — copper.hat, Oct 07 '13 at 18:54

score 1 · Answer 1 · answered Oct 07 '13 at 18:49

Look at the definition of measurability you learned. You probably proved a whole list of equivalence conditions for measurability. Of these, one is going to be particularly easy to establish for a decreasing function. Try to think geometrically about each measurability criterion and think about it in the case of decreasing functions.

Show that every decreasing function $f$: $\mathbb{R} \rightarrow \mathbb{R}$ is $\mathcal{B}(\mathbb{R})$ measurable

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