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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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&#039;t know where to start. Any help welcome.

Cahokiavv · Accepted Answer

Notice that            1              [      x      ,      ∞      )        (  y  )  =      {                            1                          y          ≥          x                                      0                          y          &amp;lt;          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.

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

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