Weak law does not imply strong law

The recursion theorem: Let $(N,0,\sigma )$ be a Peano system (def-peano-system), in particular the natural numbers $\mathbb{N}$ (def-natural-numbers). For any set $A$, any element $a\in A$, and any function $f:A\to A$, there is a unique function $g:N\to A$ such that $g(0)=a$ and $g(\sigma (n))=f(g(n))$ for all $n\in N$.

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: Let $n\ge 1$, assume the Axiom of Countable Choice (def-countable-choice), and let $a_i\le b_i$ be reals for $i<n$. Write

$R^{\circ }:=\{x\in {\mathbb{R}}^n:a_i<x_i<b_i\text{ for every }i<n\},\bar{R}:=[a,b]=\{x\in {\mathbb{R}}^n:a_i\le x_i\le b_i\text{ for every }i<n\}$

(def-multidimensional-rectangle-and-volume). Then $R^{\circ }$ is open and $\bar{R}$ is closed, so both are Borel and Lebesgue measurable, and every set $R$ with $R^{\circ }\subseteq R\subseteq \bar{R}$ is Lebesgue measurable with

${\lambda }_n(R)={\prod }_{i<n}(b_i-a_i)\mathrm{.}$

In particular this covers the four one-dimensional face conventions in each coordinate — the open box, the closed box $[a,b]$, the half-open box $B(a,b)={\prod }_{i<n}(a_i,b_i]$ of def-half-open-box, and every mixture of them, in any combination of coordinates — and it gives measure $0$ to all of them whenever $a_i=b_i$ for some $i<n$. For a half-open box with infinite parameters the value is already ${\lambda }_n(B)=\mathrm{vol}(B)$ (thm-lebesgue-measure-is-a-complete-measure).

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $\nu$ on $(S,\mathrm{\Sigma })$ is the common law of a countable independent family of $S$-valued random elements.

Measurable coordinatewise functions preserve independence: Let $(X_i{)}_{i\in I}$ be an independent family of random elements $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$. For each $i$, let $g_i:(S_i,{\mathrm{\Sigma }}_i)\to (T_i,{\mathcal{T}}_i)$ be measurable. Then the family $(g_i\circ X_i{)}_{i\in I}$ is independent.

Second Borel-Cantelli lemma under pairwise independence: Let $(A_n{)}_{n\in \mathbb{N}}$ be pairwise independent events with ${\sum }_{n=0}^{\mathrm{\infty }}\mathbb{P}(A_n)=+\mathrm{\infty }\mathrm{.}$ Then $\mathbb{P}(A_n\mathrm{i}\mathrm{.}\mathrm{o}\mathrm{.})=1.$