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Let $f$ be a continuous function defined on $E$. Is it true that $f^{-1} ( A )$ is always measurable if $A$ is measurable?

I had no clue where to start this proof from. Can someone help me out?

E W H Lee
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J.R.
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  • What is $E$? What do you mean by "measurable"? Lebesgue measurable or Borel measurable? – PhoemueX Oct 11 '14 at 04:41
  • Semi-related http://math.stackexchange.com/questions/479441/example-of-a-continuous-function-that-is-not-measurable , you should add some more information to the problem. – aram Oct 11 '14 at 09:52

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The definition of a measurable function is that the preimages of measurable sets in the range are measurable in $E$. So this question is asking if every continuous function is measurable. Assuming that the sigma algebra on the range is the one generated by open sets (i.e. the Borel algebra), then yes. The answers to " Prove Continuous functions are borel functions", should do it, although you'll have to adapt it to a more general domain.

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GrayOnGray
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