Let f be a (Lebesgue) measurable function defined on R n . Given a vector...

Eden Solomon

Eden Solomon

Answered

2022-06-24

Let f be a (Lebesgue) measurable function defined on R n . Given a vector x 0 in R n , I would like to know whether the function f ( x + x 0 ) is measurable or not. I know Φ g is measurable whenever Φ is continuous and g is measurable, and a book warns me of an example of a measurable function g and a continuous function Φ such that g Φ is not measurable. However, I have no idea how to prove or disprove measurability of f ( x + x 0 ). Can someone please give me a hand? Thank you very much.

Answer & Explanation

Quinn Everett

Quinn Everett

Expert

2022-06-25Added 23 answers

Let g ( x ) = x + x 0 . E := f 1 ( ( a , ) ) is a measurable set by definition, and g 1 ( E ) = { x x 0 : x E }, i.e. E x 0 . But the translation of a Lebesgue measurable set is Lebesgue measurable, so f g is measurable.
Hailie Blevins

Hailie Blevins

Expert

2022-06-26Added 8 answers

Thank you, and I guess you have used ( f g ) 1 ( ( a , ) ) = E x 0

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