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MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document I participate the stochastic course and we now speak about summable families. There we have the following definition:

Let $\mathrm{\Omega }$ be countable and $a:\mathrm{\Omega }\to {\mathbb{R}}_{+}\cup \{\mathrm{\infty }\}$ be a map. Then we define
$\sum _{\mathrm{\Omega }}a(\omega ):=\underset{F\subset \mathrm{\Omega },|F|<\mathrm{\infty }}{sup}\sum _{F}a(\omega )$

Now our Prof said that we can consider $\sum _{\mathrm{\Omega }}a(\omega )$ as the integral of math xmlns="http://www.w3.org/1998/Math/MathML"> a over $\mathrm{\Omega }$ to get a better connetion to measure theory afterwards when we speak about Fatou's lemma, Beppo Levi theorem ect. Because all this theorems we have seen with integrals last semester.
But somehow I don't see why this sum is equal to the integral. So I know from measure theory that if we have a simple function $f:\mathrm{\Omega }\to {\mathbb{R}}_{+}\cup \{\mathrm{\infty }\}$ where $f(\mathrm{\Omega })=\{b_1,...,b_n\}$ finately many then
${\int }_{\mathrm{\Omega }}f\mathsf{d}\mu =\sum _{i=1}^{n}f(b_i)\cdot \mu ({\mathrm{\Omega }}_i)$
where ${\mathrm{\Omega }}_i=f^{-1}(\{b_i\})$. So but here I don't think that this has to do something with simple functions right?
Therefore I wanted to ask you if someone could explain me why we can see this sum as the integral of $a$ over $\mathrm{\Omega }$.