I have a random variable ξ : Ω → R which distribution function has a density, so by definition I have that the probability measure of each ( − ∞ , x ] can be calculated by: F ξ ( x ) = P ξ ( − ∞ , x ] = ∫ − ∞ x f ξ ( y ) d y ( 1 )where the integral above is in the Lebesgue sense, with respect to the Lebesgue measure in R .My book says that a wider formula holds, that is: P ξ ( B ) = ∫ B f ξ d x , ∀ B ∈ B ( R )How can I use (1) in order to obtain this last formula? In other words, how can I extend (1) to every Borel set?

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I have a random variable   ξ  :  Ω  →      R   which distribution function has a density, so by definition I have that the probability measure of each   (  −  ∞  ,  x  ] can be calculated by:      F    ξ    (  x  )  =      P    ξ    (  −  ∞  ,  x  ]  =      ∫          −      ∞        x        f    ξ    (  y  )      d    y    (  1  )where the integral above is in the Lebesgue sense, with respect to the Lebesgue measure in       R  .My book says that a wider formula holds, that is:      P    ξ    (  B  )  =      ∫    B        f    ξ        d    x  ,    ∀  B  ∈      B    (      R    )How can I use (1) in order to obtain this last formula? In other words, how can I extend (1) to every Borel set?

gsoublogpycv8 · Accepted Answer

If suffices to show that the equality holds for sets of the form   {  (  a  ,  b  ]  :  a  &amp;lt;  b  }. Indeed,            P              ξ        (  (  a  ,  b  ]  )  =            P              ξ        (  (  −  ∞  ,  b  ]  )  −            P              ξ        (  (  −  ∞  ,  a  ]  )  =      ∫    a    b        f          ξ        (  y  )    d  y  .Now, let       C    :=  {  B  ∈      B    (      R    )  :            P              ξ        (  B  )  =      ∫    B        f          ξ        (  y  )    d  y  }. It is a monotone class that contains the above-mentioned sets. Thus,       C    =  σ  (  {  (  a  ,  b  ]  :  a  &amp;lt;  b  }  )  =      B    (      R    )

I have a random variable ξ : <mi mathvariant="normal">Ω

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