Proof of Independence is Preserved by Limits in Probability

Step 0 (An almost surely convergent subsequence). Choose a strictly increasing sequence $(k_j)_{j\in\mathbb{N}}$ such that for every $j$ and every $i\in\{1,\dots,p\}$,

$$P\bigl(|X^{k_j}_i-X_i|>2^{-j}\bigr)\le 2^{-j};$$

this is possible because, for each of the finitely many $i$, convergence in probability provides a threshold beyond which $P(|X^{k}_i-X_i|\ge2^{-j})\le2^{-j}$, and $\{|X^k_i-X_i|>2^{-j}\}\subseteq\{|X^k_i-X_i|\ge2^{-j}\}$. For fixed $i$, the events $A^i_j=\{|X^{k_j}_i-X_i|>2^{-j}\}$ satisfy $\sum_j P(A^i_j)<\infty$, so by the first Borel-Cantelli lemma $P(\limsup_j A^i_j)=0$. Off $\limsup_j A^i_j$, one has $|X^{k_j}_i-X_i|\le2^{-j}$ for all large $j$, hence $X^{k_j}_i\to X_i$ in the sense of Limit of a Sequence of Real Numbers. Let $\Omega_0=\bigcap_{i=1}^{p}(\Omega\setminus\limsup_j A^i_j)$; then $P(\Omega_0)=1$ (finite intersection of events of probability one) and on $\Omega_0$ all $p$ convergences hold.

Step 1 (Factorization over bounded continuous functions). Let $f_1,\dots,f_p:\mathbb{R}\to\mathbb{R}$ be bounded continuous functions, say $|f_i|\le C_i$. Continuous functions are measurable for the Borel $\sigma$-algebra (preimages of open sets are open, and open sets generate the Borel $\sigma$-algebra), so each $f_i(X^k_i)$ is a random variable with $\sigma(f_i(X^k_i))\subseteq\sigma(X^k_i)$, since $(f_i\circ X^k_i)^{-1}(B)=(X^k_i)^{-1}(f_i^{-1}(B))$ (elementary set algebra) with $f_i^{-1}(B)$ Borel, and $\sigma(X^k_i)=\{(X^k_i)^{-1}(B'):B'\ \text{Borel}\}$ by Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras.

Fix $k$ and abbreviate $Y_i=f_i(X^k_i)$. We claim $\mathbb{E}[Y_1\cdots Y_p]=\mathbb{E}[Y_1]\cdots\mathbb{E}[Y_p]$, by induction on $p$. For $p=1$ the claim is trivial. Bounded random variables are integrable on a probability space (dominated by an integrable constant, using monotonicity from Linearity and Monotonicity of the Lebesgue Integral). For the induction step with $p\ge2$, the grouping lemma applied to the independent family $(X^k_1,\dots,X^k_p)$ with the nonempty disjoint blocks $\{1\}$ and $\{2,\dots,p\}$ shows that $Y_1$ (which is $\sigma(X^k_1)$-measurable) and $Y_2\cdots Y_p$ (which is $\sigma(X^k_2,\dots,X^k_p)$-measurable, products of measurable functions being measurable by the argument recorded in the preliminaries of Square-Integrable Random Variables and the Mean-Square Inner Product) are independent. Both are bounded, hence integrable, so Expectation of a Product of Independent Random Variables gives $\mathbb{E}[Y_1(Y_2\cdots Y_p)]=\mathbb{E}[Y_1]\,\mathbb{E}[Y_2\cdots Y_p]$, and the induction hypothesis finishes the claim.

Now pass to the limit along $(k_j)$. Define

$$\tilde g_j=\mathbf{1}_{\Omega_0}\prod_{i=1}^{p}f_i(X^{k_j}_i)+\mathbf{1}_{\Omega\setminus\Omega_0}\prod_{i=1}^{p}f_i(X_i),$$

a random variable with $|\tilde g_j|\le C_1\cdots C_p$ everywhere. On $\Omega_0$, continuity of each $f_i$ gives $f_i(X^{k_j}_i)\to f_i(X_i)$, and products of convergent real sequences converge to the product of the limits; off $\Omega_0$ the sequence is constant. Hence $\tilde g_j\to\prod_i f_i(X_i)$ at every point of $\Omega$, and the dominated convergence theorem (with the constant dominating function) yields $\mathbb{E}[\tilde g_j]\to\mathbb{E}[\prod_i f_i(X_i)]$. Moreover $\bigl|\mathbb{E}[\tilde g_j]-\mathbb{E}[\prod_i f_i(X^{k_j}_i)]\bigr|\le 2C_1\cdots C_p\,P(\Omega\setminus\Omega_0)=0$. The same argument applied to a single factor gives $\mathbb{E}[f_i(X^{k_j}_i)]\to\mathbb{E}[f_i(X_i)]$. Combining with the stage-$k_j$ factorization:

$$\mathbb{E}\Bigl[\prod_{i=1}^{p}f_i(X_i)\Bigr]=\lim_j\prod_{i=1}^{p}\mathbb{E}[f_i(X^{k_j}_i)]=\prod_{i=1}^{p}\mathbb{E}[f_i(X_i)].$$

Step 2 (Half-line factorization). Fix $a_1,\dots,a_p\in\mathbb{R}$. For $m\ge1$ let $f^m_{a}(x)=1$ for $x\le a$, $f^m_a(x)=1-m(x-a)$ for $a<x<a+1/m$, and $f^m_a(x)=0$ for $x\ge a+1/m$; each $f^m_a$ is continuous and bounded by $1$. As $m\to\infty$, $f^m_a(x)\to\mathbf{1}_{(-\infty,a]}(x)$ at every $x$. Applying Step 1 with $f_i=f^m_{a_i}$ and letting $m\to\infty$ with the dominated convergence theorem (dominating constant $1$; the product of the indicators is the indicator of the intersection) gives

$$P\bigl(X_1\le a_1,\dots,X_p\le a_p\bigr)=\prod_{i=1}^{p}P(X_i\le a_i)\qquad\text{for all }a_1,\dots,a_p\in\mathbb{R}. \tag{*}$$

Step 3 (From half-lines to Borel sets). We show by induction on $r\in\{0,1,\dots,p\}$ the claim $C_r$: for all Borel sets $B_1,\dots,B_r$ and all reals $a_{r+1},\dots,a_p$,

$$P\Bigl(\bigcap_{i\le r}\{X_i\in B_i\}\cap\bigcap_{i>r}\{X_i\le a_i\}\Bigr)=\prod_{i\le r}P(X_i\in B_i)\prod_{i>r}P(X_i\le a_i).$$

$C_0$ is (*). Assume $C_{r-1}$ and fix Borel $B_1,\dots,B_{r-1}$ and reals $a_{r+1},\dots,a_p$. Let

$$\Lambda=\Bigl\{B\in\mathcal{B}(\mathbb{R}):P\Bigl(\{X_r\in B\}\cap D\Bigr)=P(X_r\in B)\cdot c\Bigr\},$$

where $D=\bigcap_{i<r}\{X_i\in B_i\}\cap\bigcap_{i>r}\{X_i\le a_i\}$ and $c=\prod_{i<r}P(X_i\in B_i)\prod_{i>r}P(X_i\le a_i)$. By $C_{r-1}$, $\Lambda$ contains every half-line $(-\infty,a]$; it contains $\mathbb{R}$ by letting $a\to\infty$ along $(-\infty,n]\uparrow\mathbb{R}$ and using continuity of $P$ from below (countable additivity) on both sides, which yields $P(D)=c$. Both $B\mapsto P(\{X_r\in B\}\cap D)$ and $B\mapsto P(X_r\in B)\,c$ are finitely additive and continuous from below in $B$, so $\Lambda$ is closed under proper differences and increasing countable unions; together with $\mathbb{R}\in\Lambda$ this makes $\Lambda$ a $\lambda$-system in the sense of Dynkin's Pi-Lambda Theorem. The half-lines form a $\pi$-system generating $\mathcal{B}(\mathbb{R})$ (every open interval is a countable combination of half-lines, $(a,b)=\bigcup_{n\ge1}\bigl((-\infty,b-(b-a)/(2n)]\setminus(-\infty,a]\bigr)$; every open set is a countable union of open intervals with rational endpoints by the density of the rationals; and open sets generate the Borel $\sigma$-algebra by Borel Sigma-Algebra on the Real Line). By Dynkin's theorem, $\Lambda=\mathcal{B}(\mathbb{R})$, which is $C_r$.

Step 4 (Conclusion). $C_p$ states that $P\bigl(\bigcap_{i=1}^{p}\{X_i\in B_i\}\bigr)=\prod_{i=1}^{p}P(X_i\in B_i)$ for all Borel $B_1,\dots,B_p$. For any nonempty subfamily of indices, choose $B_i=\mathbb{R}$ for the omitted indices; the corresponding factors equal $1$, so the product identity holds for every subfamily. By Independence of Events and of Random Variables (see also the discussion in Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras of inserting $\Omega$ for omitted factors), the random variables $X_1,\dots,X_p$ are independent. $\square$