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I guess this is a trivial problem. I was reading about expected value on wiki and I came across a notation of an integral I don't understand. There is a statement that a general case of expected value has this form:

$$E[X]=\int_{\Omega} X(\omega)\,dP(\omega)$$

with a comment that this is a Lebesgue integral. I was taught to calculate integrals or multi integrals with respect to a number of variables, not functions. When I see the term $dP(\omega)$, I am confused!

I know an expected value can also be expressed in this form

$$E[X] = \int_{X} x\,p(x)\,dx$$

because it is simply a weighted sum / integral of a random variable over probabilities associated with its realizations.

How to understand an integral when it is calculated with respect to a function?

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Formally speaking, a probability space is a measure space. Given a set of possible outcomes, we can find the measure of that set - the probability of being in that set.

That integral "$dP(\omega)$" is simply the integral with respect to that measure.

In practice, how will we evaluate it? We'll find a density function $\rho$, or a probability mass function $p$, and convert it to something like $$E(X) = \int_{\Omega} x\rho(x)\,dx$$ in the density case (where $\Omega$ is the space of possible values), or $$E(X) = \sum_{x\in \Omega}xp(x)$$ Writing it in terms of the probability measure $P(\omega)$ allows us to unify those two expressions, as well as more complicated cases (that hardly ever come up in practice).

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  • $\begingroup$ I am afraid don't know what a measure or a measure of set is but when I read your explanation I kind of understand that $P(\omega)$ is one of the measures we could use. In this example, it is Probability but it could be something esle. When we integrate with respect to $dP(\omega)$, we take numbers a function returns (probability) and not what variables are. Could integration be considered as a operand over product of terms and this time the "term" is not a variable but a value of a function? I am confused :p but thanks for the answer. $\endgroup$
    – Celdor
    Commented Mar 12, 2019 at 12:04
  • $\begingroup$ At your level of understanding? Ignore that talk about measures. We know how to find the expected value if it's a "continuous" random variable with a density function (the integral expression), and we know how to find the expected value if it's a discrete random variable with a probability mass function (the sum expression). That's all you really need. Later, when you've got more of a real analysis background, you can come back and update the theory behind things with the idea of measures. $\endgroup$
    – jmerry
    Commented Mar 12, 2019 at 12:22
  • $\begingroup$ Thanks for your time. I am not going to ignore it as I figured out my question is also about that part which as it turned out is a measure. This is what confuses me. Cheers! $\endgroup$
    – Celdor
    Commented Mar 12, 2019 at 12:30

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