How to show that ∫ ‖ y − y 0 ‖ ν ( d y ) &lt; ∞ ?Let X be a uniformly random variable on [0,1]. Since Y is supported on [cos(2),1], then we can define a probability measure on R so that Y ∼ νHow to show that for any y 0 ∈ R we have ∫ ‖ y − y 0 ‖ ν ( d y ) &lt; ∞ ?

Question

How to show that   ∫  ‖  y  −      y    0    ‖  ν  (  d  y  )  &amp;lt;  ∞  ?Let X be a uniformly random variable on [0,1]. Since Y is supported on [cos(2),1], then we can define a probability measure on R so that   Y  ∼  νHow to show that for any       y    0    ∈  R we have  ∫  ‖  y  −      y    0    ‖  ν  (  d  y  )  &amp;lt;  ∞  ?

Bobby Mcconnell · Accepted Answer

Step 1From the definitions you have that                    (1)                              ∫                                    R                                                |                y        −                  y          0                          |                ν        (        d        y        )        =                  ∫                      Ω                                    |                Y        −                  y          0                          |                        d        P        ⩽                  ∫                      Ω                          (                  |                Y                  |                +                  |                          y          0                          |                )                d        P        ⩽        1        +                  |                          y          0                          |                    as       |    Y      |    ⩽  1. For a more elementary approach note that                    (2)                              ∫                                    R                                                |                y        −                  y          0                          |                ν        (        d        y        )        ⩽                  ∫                                    R                                                |                y                  |                ν        (        d        y        )        +                  ∫                                    R                                                |                          y          0                          |                ν        (        d        y        )            where I used the triangle inequality and the linearity of the integral. Now, the first integral in the RHS of (2) is just the expected value of |Y|, what is finite because       |    Y      |    ⩽  1 so                    (3)                              ∫                                    R                                                |                y                  |                ν        (        d        y        )        ⩽                  ∫                      [            −            1            ,            1            ]                          ν        (        d        y        )        =        Pr        [                  |                Y                  |                ⩽        1        ]        =        1            Step 2Now the second integral in the RHS of (2) is just       |        y    0        |   by the linearity of the integral, so again we reach the same bound as in (1).

How to show that int ∥y-y_0∥ν(dy)<infty?

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