I met an excercise in the book by Rabi Bhattacharya and Edward C. Waymire. Suppose that μ , ν are probbaility measures on R d , with ν absolutely continuous with pdf f, i.e., d ν = f ( x ) d x. How to show that the convolution, μ ∗ ν, is also absolutely continuous? Thanks!

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I met an excercise in the book by Rabi Bhattacharya and Edward C. Waymire. Suppose that   μ  ,  ν are probbaility measures on             R        d  , with   ν absolutely continuous with pdf   f, i.e.,   d  ν  =  f  (  x  )  d  x. How to show that the convolution,   μ  ∗  ν, is also absolutely continuous? Thanks!

coquinarq1 · Accepted Answer

HintIf      A    is a measurable subset of                 R        d      , then                    μ        ∗        ν        (        A        )                            =                  ∫                                                    R                            d                                                ∫                      A            −            y                          d        ν        (        x        )        d        μ        (        y        )                                          =                  ∫                                                    R                            d                                                ∫                      A            −            y                          f        (        x        )        d        x        d        μ        (        y        )                 .            If      A    has Lebesgue measure 0 what is the value of this integral?

Pitrellais · Accepted Answer

pdf is                              ∫                                                    R                            d                                                ∫                      A            −            y                          f        (        x        )        d        x        d        μ        (        y        )                            =                  ∫                                                    R                            d                                                ∫          A                f        (        x        −        y        )        d        x        d        μ        (        y        )                                          =                  ∫          A                          ∫                                                    R                            d                                      f        (        x        −        y        )        d        μ        (        y        )        d        x                 ,            which gives      ∫                            R                d              f  (  x  −  y  )  d  μ  (  y  )as the density of      μ  ∗  ν

I met an excercise in the book by Rabi Bhattacharya and Edward C. Waymire. Suppose that μ

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