Series of independent random variables are independent

In the proof of a theorem my lecturer seemed to have used this fact without first proving it: Let $(X_i)_{i \geq 1}$ be real-valued independent random variables on $(\Omega,\mathscr{F},\mathbb{P})$, and let $\mathbb{N}$ be partitioned into disjoint countable sets $(I_k)_{k\geq 1}$. If $Y_k:=\sum_{i \in I_k} X_i$ converges for all $k$, then $(Y_k)_{k\geq 1}$ are independent random variables. (It is clear to me that each $Y_k$ is a random variable, but I need to prove independence.)

Specifically, the proof uses this fact with $X_i =$ the $i$-th Rademacher function, suitably scaled.

Can somebody give a hint please?

Edit Independence of random variables is defined in terms of the corresponding $\sigma$-algebras on $\Omega$. For real-valued random variables, I understand that $(Y_k)_{k\geq1}$ are independent if and only if $\mathbb{P}(Y_1 \leq y_1, \dotsc, Y_k \leq y_k) = \prod_{1\leq j \leq k}\mathbb{P}(Y_j \leq y_j)$ for all $k$.

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How you have defined independence for a sequence of random variables $\{X_i\}_{i=1}^{\infty}$ in your class at this point? – Tom Nov 7 '13 at 18:03
@Tom Please see the edit! – user71815 Nov 7 '13 at 18:11

We have to show that for each integer $N$, $(Y_j,1\leqslant j\leqslant N)$ are independent random variables. So take $B_j,1\leqslant j\leqslant N$ some Borel sets. We have $\{Y_j\in B_j\}\in \sigma(X_i,i\in I_j)=:\mathcal F_j$, so we have to show that the $\sigma$-algebras $(\mathcal F_j,1\leqslant j\leqslant N)$ are independent.

By an approximation argument, it's enough to see that $\left(\bigcup_{l=1}^\infty\mathcal F_j^{(l)},1\leqslant j\leqslant N\right)$ are independent, where $\mathcal F_j^{(l)}:=\sigma(X_i,i\in I_j\cap [-l,l])$. Indeed, for each $j$, $\mathcal F_j$ is generated by the algebra $\bigcup_{l=1}^\infty\mathcal F_j^{(l)}$.

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Can you elaborate on "$I_j \cap [-l,l]$" please? $I_j$ is a subset of $\mathbb{N}$, so $\mathcal{F}_j^{(1)} \subseteq \mathcal{F}_j^{(2)} \subseteq \dotsb$, giving $\bigcup_{l=1}^\infty \mathcal{F}_j^{(l)} = \mathcal{F}_j$? Do you mean we should show $(\mathcal{F}_j^{(l)}, 1\leq j \leq N, l \in \mathbb{N})$ are independent? – user71815 Nov 8 '13 at 15:52
The union is not necessarily $\cal F_j$, but it is an algebra which generates this $\sigma$-algebra. For your second question, yes. – Davide Giraudo Nov 8 '13 at 15:58
Okay, would you mind editing your answer? – user71815 Nov 8 '13 at 16:00
I made a small edit. Do you need more details? – Davide Giraudo Nov 8 '13 at 21:05
You have basically reduced the question to "If $X_1,X_2,X_3$ are independent random variables, then $\sigma(X_1,X_2)$ and $\sigma(X_3)$ are independent", am I right? I'll need to find a proof of this... – user71815 Nov 8 '13 at 22:25