I have a question about a finite σ measure on ( <mrow class="MJX-TeXAtom-ORD"> <

opepayflarpws 2022-06-25 Answered
I have a question about a finite σ measure on ( R + , B ( R + ) ):
S h o w   t h a t   2 R + x μ ( [ x , [ ) d x = R + x 2 μ ( d x )
I know that I should use fubini, but unfortunately I don't know where to start. Any help welcome.
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Answers (1)

Cahokiavv
Answered 2022-06-26 Author has 31 answers
Notice that
1 [ x , ) ( y ) = { 1 y x 0 y < x = 1 ( 0 , y ] ( x )
which implies:
R + x 1 [ x , ) ( y ) d x = ( 0 , y ] x d x = y 2 2
So
R + x μ ( [ x , ) ) d x = R + R + x 1 [ x , ) ( y ) μ ( d y ) d x = = ( 1 ) R + R + x 1 [ x , ) ( y ) d x μ ( d y ) = = R + y 2 2 μ ( d y )
where (1) is by Fubini-Tonelli.

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