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I'm, of course, familiar with the definition that a function f:R->R is Lebesgue measurable if the preimage of intervals/open sets/closed sets/Borel sets is Lebesgue measurable.

We also say a function between general measurable X, Y spaces is measurable if the preimage of a set in the sigma algebra corresponding to Y is in the sigma algebra corresponding to X.

For a Lebesgue measurable function f:R->R is it necessarily true that the preimage of a Lebesgue measurable set is Lebesgue measurable? I don't see that this need necessarily be the case.

Am I correct in thinking that a Lebesgue measurable function f:R->R is a measurable function from R with the Lebesgue sigma algebra to R with the Borel sigma algebra (and NOT the Lebesgue sigma algebra)?

I'd be very grateful if anyone could clear up my confusion.