Does a strong law of large numbers hold in the continuum?

Question

Can we construct an i.i.d. family of Rademacher random variables $(X_t)_{t\in\mathbb{R}}$ defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ (so, in particular, $\forall t \in \mathbb{R}, \forall i\in\{-1,+1\}, \mathbb{P}[X_t=i]=\frac{1}{2}$) such that for $\mathbb{P}$-a.s. $\omega \in \Omega$ we have that $$\mathbb{R} \to \{-1,+1\}, t\mapsto X_t(\omega)$$ is measurable? If so, and $-\infty<a<b<+\infty$, what could be said of the random variable $$\omega\mapsto \frac{1}{b-a}\int_{[a,b]}X_t(\omega)\operatorname{d}t?$$ Does it satisfy some kind of strong law of large numbers? Intuitively - and here my intuition could be very wrong - this integral "should be" just a series of infinitesimal i.i.d. random variables of zero mean...

Answer 1

I will show that joint measurability is impossible. That is, you cannot cook up such a probability space with $(\omega,t) \mapsto X_{t}(\omega)$ jointly measurable (that is, as a map from $\Omega \times \mathbb{R}$ with product of $\mathcal{F}$ and Lebesgue to $\mathbb{R}$.) This seems to be a stronger assumption than what you asked for. At the same time, it is somewhat awkward to say "$t \mapsto X_{t}(\omega)$ is measurable for $\mathbb{P}$-a.e. $\omega \in \Omega$" since it's not clear that "is measurable" will be in $\mathcal{F}$ or how to build such a $\Omega$.

Given $a,b \in \mathbb{Q}$, define $Y_{a,b} = \int_{a}^{b} X_{s} \, ds$. Notice that Fubini's Theorem implies \begin{equation*} \mathbb{E}(Y_{a,b}^{2}) = \int_{a}^{b} \int_{a}^{b} \mathbb{E}(X_{s} X_{\xi}) \, ds \, d \xi = 0. \end{equation*} Therefore, $Y_{a,b} = 0$ $\mathbb{P}$-a.s. $\mathbb{Q} \times \mathbb{Q}$ is countable so, in fact, $\{Y_{a,b} \, \mid \, a,b \in \mathbb{Q}\} = \{0\}$ almost surely. Since $t \mapsto X_{t}$ is measurable in this event, we are left to conclude that $X \equiv 0$ almost surely. (A bounded, measurable function whose integral vanishes in every interval with rational endpoints equals zero almost surely.) This contradicts your specification that $\{X_{t}\}_{t \in \mathbb{R}} \subseteq \{-1,1\}$.

The argument is taken directly from Chapter 1 of Revuz and Yor (see Section 3).