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.

Question

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.

Quinn Everett · Accepted Answer

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 · Accepted Answer

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