I was reading the proof given by proofwiki on the continuity of probability measures (here).
I can follow the proof but my issue is on the following step:
So I can understand that we have $$\text{Pr}(A) = \text{Pr}(A_0 \cup B_1 \cup \cdots ) = \text{Pr}(A) + \sum_{i>0} \text{Pr}(B_i)$$ Because the events $B_i$ are disjoint and so by the axiom of countable additivity we can do this. However, where does the limit come from? Proofwiki justifies everything saying that it comes from the definition of probability measure, but it doesn't because if you look at its definition, a probability measure is just a measure that is normalized with respect to the universe set $\Omega$. But this is the definition of a measure:
Let $(X, \Sigma)$ be a measurable space. Let $\mu: \Sigma \longrightarrow \overline{\mathbb{R}}$ be a mapping. Here $\overline{\mathbb{R}}: = \mathbb{R} \cup \{+\infty, -\infty\}$ is the extended real number line. Then $\mu$ is called a measure on $\Sigma$ if and only if
- $\mu(E) \geq 0$ $\qquad \forall E\in \Sigma$ (positive)
- $\mu\left(\bigcup_{n=1}^\infty S_n\right) = \sum_{n=1}^\infty
\mu(S_n)$ for every sequence of pairwise disjoint sets
$\{S_n\}\subseteq \Sigma$ (countably additive function) - $\exists E\in \Sigma : \,\, \mu(E)$ is finite. (or $\mu(\emptyset) =
0$)
Where does the limit come from?
Best Answer
I think you're going too deep: the definition of an infinite sum is exactly the limit of partial sums. $$\sum_{i > 0} Pr(B_i)$$ literally is defined as $\lim_{n \to\infty} \sum_{i = 1}^n Pr(B_i)$.