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Borel sets are defined because we want a measurable set of sets, and we want them to have nice topological properties.

.... Uhm.... but when do we actually take the size of Borel sets? When are they ever measured?

In Probability Theory, we measure $X \in A$, where $A$ is a Borel set. But we are not measuring a Borel set, since $X \in A$ is not a Borel set. It lies in the abstract sigma-algebra $\mathcal{F}$.

So, when do we actually measure a Borel set?

Cinad
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    ... as soon as we pick a (Borel) measure – Hagen von Eitzen Oct 12 '17 at 20:08
  • All the time. Many functions of interest can be approximately by simple functions of the form $$f = c_1 \textrm{Char}(E_1) + \cdots + c_t \textrm{Char}(E_t)$$

    for Borel sets $E_1, ... , E_t$. The integral of such a function with respect to a Borel measure will be $$c_1 \textrm{meas}(E_1) + \cdots + c_t \textrm{meas}(E_t)$$

     – D_S Oct 12 '17 at 20:09
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    "In probability theory, we measure $X\in A$ where $A$ is a Borel set...." You probably mean: we measure ${X\in A}={\omega\in\Omega\mid X(\omega)\in A}$ where $X$ denotes a random variable. That gives a measure on Borel-sets: the Borel-set $A$ gets measure $P(X\in A)$. Every random variable induces a measure on Borel-sets like that. – drhab Oct 12 '17 at 20:15

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