probability of a limiting sum Suppose that $U_i$ are uniformly distributed on (0,1) and are independent. For all possible increasing index sets comprising the family $J$, I am trying to show that
$P(\cap_{j\in J} \{\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^n X_{j_k}}{n}\} = \frac{1}{2}) = 0.$
My attempt is as follows: denote by $A_i = \{\omega \in \Omega:\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^n X_{i_k}}{n} = \frac{1}{2}\}$.
Then instead of showing that $P(\cap A_i) = 0,$ I can show that $P(\cup A_i^c) = 1.$
However, $P(\cup A_i^c) \le \sum P(A_i^c)$.
Now, there's a theorem which says that if $U_i$'s are iid, $E[U_i] = \mu$ and $E[U_i^4] < \infty$ then $S_n = X_1 + \dots + X_n$ has the property that $S_n/n \rightarrow \mu$ a.s.
In the above case, $P(A_i) = 1$ because of the above theorem implying that $P(A_i^c) = 0$. This contradicts what I am supposed to prove.
Any thoughts or hints? One hint given is to use the fact that $\{U_1,U_2,\dots\}$ are dense in $(0,1)$ save for a set of measure 0. But I'm stuck.

The idea is simpler to explain when $(X_i)$ is i.i.d. Bernoulli with $P(X_i=0)=P(X_i=1)=\frac12$. For every $j$ in $J$, let $S_n^j=X_{j_1}+X_{j_2}+\cdots+X_{j_n}$ and $$A_j=\{\lim\limits_{n}\tfrac1nS_n^j=1/2\}.$$ Let $\omega$ in $\Omega$ and $Z(\omega)=\{n\mid X_n(\omega)=0\}\subseteq\mathbb N$.


*

*Either $Z(\omega)$ is infinite then $j=Z(\omega)$ yields $S_n^j(\omega)=0$ for every $n$ hence $\omega$ is not in $A_j$ because $\lim\limits_{n}\tfrac1nS_n^j(\omega)=0$.

*Or $Z(\omega)$ is finite then $j=\mathbb N$ yields $S_n^j(\omega)\geqslant n-|Z(\omega)|$ for every $n$ hence $\omega$ is not in $A_j$ because $\lim\limits_{n}\tfrac1nS_n^j(\omega)=1$.


This proves that, for every $\omega$ in $\Omega$, there exists $j$ in $J$ such that $\omega$ is not in $A_j$, that is, $$\bigcap_{j\in J}A_j=\varnothing.$$
in particular, the set on the LHS is an event and its probability is zero. (Note that the appearance of the empty set on the RHS of this identity is fortunate since one cannot know a priori that the set on the LHS, being an uncountable intersection of events, is an event.) 
Can you adapt this to the case when $(X_i)$ is i.i.d. uniform on $(0,1)$?
