MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Consider probability spaces ( &#x03A9; , G , P ) and ( &#x03A9; , F , Q ) with the properties F &#x2286; G and P | F = Q.Let X be a Q-integrable (and F -measurable) random variable. Is it true, that

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MathJax(?): Can&#039;t find handler for document  MathJax(?): Can&#039;t find handler for document  Consider probability spaces   (  &amp;#x03A9;  ,      G    ,  P  ) and   (  &amp;#x03A9;  ,      F    ,  Q  ) with the properties       F    &amp;#x2286;      G   and   P            |                      F              =  Q.Let   X be a   Q-integrable (and       F  -measurable) random variable. Is it true, that

coquinarq1 · Accepted Answer

It is true. In fact, it is true when   0  ≤  X  ≤  ∞ is   F-measurable, which then implies it is true for integrable   F-measurable   X since   X  =      X    +    −      X    −  −. To prove for   0  ≤  X  ≤  ∞, you can prove it for simple functions on   F and then take a sequence of nonnegative simple functions       ϕ    n    ↗  X and use the monotone convergence theorem.

Consider probability spaces ( <mi mathvariant="normal">Ω , <mrow class

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