Quantifiers inside probabilities

Question

Let $ Y $ be a random vector. If $ P\left(\lim_{n\to\infty}Y_{n} = Y\right) = 1 $ we say, that $ Y_{n} $ converges almost surely to $ Y $, written $ Y_{n}\xrightarrow{a.s. \ P} Y $.

Statement:

It holds that $ Y_{n}\xrightarrow{a.s. \ P} Y $ if and only if for every $ \epsilon > 0 $, $$ P\left(\| Y_{k}-Y \| < \epsilon, \forall k\geq n \right) \to 1 \quad \text{as}\quad n\to\infty.$$

Proof:

Let $ A_{n,\epsilon} = \left\lbrace \|Y_{k}-Y\| < \epsilon, \forall k\geq n \right\rbrace$ and assume that $ Y_{n}\xrightarrow{a.s. \ P} Y $. Then \begin{align} &P\left(\lim_{n\to\infty}Y_{n} = Y\right)\notag\\ =& P\left( \forall \epsilon > 0, \exists n \ \text{such that } \|Y_{k}-Y\|<\epsilon, \forall k\geq n \right)\notag\\ =&P\left( \bigcap_{\epsilon > 0}\bigcup_{n} A_{n,\epsilon} \right) = 1. \end{align} As $ \epsilon $ tends to zero, the set $ \cup_{n}A_{n,\epsilon} $ decreases to $ \cap_{\epsilon}\cup_{n}A_{n,\epsilon} $ and hence $P\left( \bigcap_{\epsilon > 0}\bigcup_{n} A_{n,\epsilon} \right) = 1$ is equivalent to $ P\left(\cup_{n} A_{n,\epsilon} \right) = 1, $ for all $ \epsilon > 0 $.

Then, since $ A_{n} $ is an increasing sequence of sets such that $$ \lim_{n\to\infty}A_{n,\epsilon} = \bigcup_{n} A_{n,\epsilon} $$ it follows by continuity of probability measures, that $$ P\left( A_{n,\epsilon} \right) \to 1 \quad \text{as} \quad n\to\infty, $$ for all $ \epsilon > 0 $ which finishes the proof.

Question: How are the universal and existential quantifiers in $$P\left( \forall \epsilon > 0, \exists n \ \text{such that } \|Y_{k}-Y\|<\epsilon, \forall k\geq n \right)$$ converted to intersection and union in $$P\left( \bigcap_{\epsilon > 0}\bigcup_{n} A_{n,\epsilon} \right).$$ I failed to construct a simple example and realize this. Are there any conventions about this or is it more like a triviallyty?

Answer 1

There is a tight junction between $\forall$ and $\bigcap$ on one side and $\exists$ and $\bigcup$ on the other side, given by the definition of $\bigcap$, i.e. $$ \bigcap_i A_i = \{x \mid \forall i \in I: x \in A_i\} $$ and $$ \bigcup_i A_i = \{x \mid \exists i \in I: x \in A_i\}. $$ In your case, we consider the sets $$ A_{n,\epsilon} := \{\forall k \ge n: \|Y_k - Y\| < \epsilon \} $$ For each $\omega \in \Omega$ we have by the above \begin{align*} \forall \epsilon \,\exists n \, \omega \in A_{n,\epsilon} &\iff \forall\epsilon\, \omega \in \bigcup_n A_{n,\epsilon}\\ &\iff \omega \in \bigcap_{\epsilon > 0} \bigcup_n A_{n,\epsilon} \end{align*} Hence $$ \{\omega: \forall \epsilon \,\exists n \, \omega \in A_{n,\epsilon}\} = \bigcap_{\epsilon > 0} \bigcup_n A_{n,\epsilon} $$