[944] General Question About Measurability

[Gautam Iyer]

On Tue, Nov 07, 2017 at 08:37:50AM -0500, Richard Rosenbaum wrote:

> Say X, Y, and Z are RVs. G is a sigma algebra.
> 
> Z = X + Y
> 
> If X and Y are both G-measurable, does this imply that Z is also
> G-measurable?

Yes! Try it, it isn't too hard.

> I guess more generally - say a random variable is some function of
> other RVs. does the measurability of the two RVs that the function
> takes as inputs imply that the function itself is measurable with
> respect to that same sigma algebra?

Almost always. To be 100% correct:

    Suppose f is a function from R^2 to R (i.e. takes two real numbers
    as inputs), AND is measurable with respect to the Borel sigma
    algebra. If X and Y are any two random variables, then f(X, Y) is
    also a random variable.

It's really really hard to write down non-Borel measurable functions. So
almost every function you encouter will be Borel measurable, and so
almost every operation you do to random variables will yield a random
variable.

Best,

Gautam

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