We know that if $f$ is a continuous function between two topology space $X$ and $Y$ then $f$ is Borel measurable. And in other hand we have that composition of two measurable function is measurable and every Borel measurable function is Lebesgue measurable. So could we say composition of two continuous functions is Lebesgue measurable?
Measurability of continuous function
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analysis
1 Answers
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The composition of two continuous functions is continuous, thus Borel measurable, thus Lebesgue measurable.