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If I have two random variables X and Y, when am I allowed to multiply them? i.e. Z=XY

Let S_1 and S_1 be sigma algebras such that S_1 is contained in S_2

Cases

i) X and Y are both S_1 measurable

It seems clear that Z=XY exists and is also S_1 measurable

ii) X is S_1 measurable and Y is S_2 measurable

In this case X is also S_2 measurable, but Y is not S_1 measurable. (Am I correct to say this?)

Can we form Z=XY and if so does Z simply become S_2 measurable?

iii) Assume S_3 is not a subset of either S_1 or S_2

Can we write Z=XY?

Thanks for your help

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# Measurability of random variables

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