A really strange question here in my opinion: Let A , B be two measurable subsets of

istremage8o 2022-05-23 Answered
A really strange question here in my opinion:
Let A , B be two measurable subsets of R 1 . Define f ( x ) = | ( A x ) B | . Evaluate R 1 f d x. Here | | refers to the measure.
Here f is clearly non-negative, so I tried using the definition of Lebesgue integral:
R 1 f d x = sup { R 1 s d x : 0 s f } ,
where s are simple functions. But I realised that I couldn't come up with any simple functions.
I tried breaking up R 1 into two parts:
E 1 = { y R 1 : y A B } , E 2 = { y R 1 : y A B }
and then considering the sum
E 1 f d x + E 2 f d x
and then I'm clueless as to how to proceed. I'm guessing the answer should be something intuitive like | A B | . Any help would be appreciated.
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Answers (1)

Buckokasg
Answered 2022-05-24 Author has 7 answers
One could proceed as follows. Note that we can rewrite f ( x ) = | ( A x ) B | = R 1 B ( y ) 1 A ( x + y ) d y. Then the question becomes evaluating the expression
R R 1 B ( y ) 1 A ( x + y ) d y d x .
By translation invariance of the Lesbeque measure on R , it follows that
R R 1 B ( y ) 1 A ( x + y ) d y d x = R R 1 B ( y ) 1 A ( y ) d y d x = R 1 B ( y ) d y R 1 A ( x ) d x = | A | | B | .
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