Show that: $f : \mathbb{R} → \mathbb{R} ; \ x \mapsto \cos^2(x)$ is measurable with respect to the Lebesgue measure space

Asked Nov 16 '19 at 13:04

Active Nov 16 '19 at 13:38

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How can I show that $f : \mathbb{R} → \mathbb{R} ; \ x \mapsto \cos^2(x)$ is measurable with respect to the Lebesgue measure space?

My first thought is that it should be Borel-measurable, so if I can prove that, then it is obviously Lebesgue-measurable, but I'm not sure how to start the proof. Any help would be great, thanks.

edited Nov 16 '19 at 13:38

Noa Even

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asked Nov 16 '19 at 13:04

userpisquared

1

Every continuous function is Borel-measurable. – drhab Nov 16 '19 at 13:06
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The @drhab comment is a hint of how to start. It is probably just as easy to prove every continuous function is Borel measurable as it is to prove $\cos^2 x$ is Borel measurable. – GEdgar Nov 16 '19 at 13:19
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You can also take a look here. – drhab Nov 16 '19 at 13:21
Perfect, thanks, that should do it. – userpisquared Nov 16 '19 at 13:25

Show that: $f : \mathbb{R} → \mathbb{R} ; \ x \mapsto \cos^2(x)$ is measurable with respect to the Lebesgue measure space

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