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My professor seems to randomly consider the product $XY$ or the sum $X+Y$ of random variables $X$, $Y$. Do they always exist as a random variable, too?

Upon some research I learned that the random variables with expected value $E\left[\left|X\right|^p\right]<\infty$ form a vector-space. So the sum $X+Y$ as linear combination of random variables is a random variable. But I found no explanation for the existence of the product $XY$ (also not for $|X|$).

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    A random variable is, by definition, a measurable function defined on a probability space. It can be shown that sum and the product of measurables functions is still measurable. – 2017-02-22
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    Note that having finite expectation is _not_ inherent in the concept of "being a random variable". – 2017-02-22
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    @Augustin Yes, absolutely. But some elementary approaches to probability may sweep that fact under the carpet, trying to keep things simple. Then, to compensate, one simply has to postulate the existence of a joint probability distribution whenever two or more random variables occur together. – 2017-02-22
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    We only consider real-valued random variables. Sum and product of functions need componentwise sum and product in the range, do they not? In this case, sum and product of random variables are random variables, also if the expected value of each is infinite? – 2017-02-22

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