Can random variables that almost surely solve equations be repaired to surely solve these equations?

In Terry's answer, he shows that his original question reduces to the question of whether, given a $\sigma$-algebra $\mathcal F$ on some set $X$ and a measure $\mu$ on $(X,\mathcal F)$, there is a ``splitting'' of the quotient algebra $\mathcal F / \mathcal N$, where $\mathcal N$ denotes the ideal of $\mu$-null sets. In this context, a splitting is a Boolean homomorphism $\Phi: \mathcal F / \mathcal N \rightarrow \mathcal F$ such that $\Phi([A]) \in [A]$ for all $A \in \mathcal F$. (Some authors call this a lifting instead of a splitting.) When some such $\Phi$ exists, let us say that $(X,\mathcal F,\mu)$ has a splitting.

I did some digging on this question this afternoon, and found two very good sources of information: David Fremlin's article in the Handbook of Boolean Algebras (available here) and a survey paper by Maxim Burke entitled "Liftings for noncomplete probability spaces" (available here). I'll summarize some of what I found below to supplement what Terry mentions in his answer. He mentions already that it is independent of ZFC whether $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting:

$\bullet$ (von Neumann, 1931) Assuming $\mathsf{CH}$, $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting.

$\bullet$ (Shelah, 1983) There is a forcing extension in which $([0,1],\text{Borel},\text{Lebesgue})$ has no splitting.

Also mentioned already is the fact that if we expand the $\sigma$-algebra in question from the Borel sets to all Lebesgue-measurable sets, then the situation is more straightforward:

$\bullet$ (Maharam, 1958) If $(X,\mu)$ is a complete probability space, then $(X,\mu\text{-measurable},\mu)$ has a splitting.

Now on to some not-yet-mentioned results. First, it's worth pointing out that one can obtain splittings with nice extra properties.

$\bullet$ (Ioenescu-Tulcea, 1967) Let $G$ be a locally compact group, and let $\mu$ denote its Haar measure. Then $(G,\mu\text{-measurable},\mu)$ has a translation-invariant splitting (which means $\Phi([A+c]) = \Phi([A])+c$ for every $\mu\text{-measurable}$ set $A$).

Once again, shrinking our $\sigma$-algebra from all $\mu$-measurable sets to only the Borel sets causes problems.

$\bullet$ (Johnson, 1980) There is no translation-invariant splitting for $([0,1],\text{Borel},\text{Lebesgue})$.

Thus, interestingly, Shelah's consistency result becomes a theorem of $\mathsf{ZFC}$ if we insist on the splitting being translation-invariant (with respect to mod-$1$ addition). More generally:

$\bullet$ (Talagrand, 1982) If $G$ is a compact Abelian group and $\mu$ is its Haar measure, then there is no translation-invariant splitting for $(G,\text{Borel},\mu)$.

What stood out to me most in Fremlin and Burke's articles is how many questions seem to be wide open.

Open question: Is it consistent that every probability space has a lifting?

If yes, this would give a consistent positive answer to Terry's original question.

Open question: Is it consistent with $2^{\aleph_0} > \aleph_2$ that $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting?

(Carlson showed that it is consistent to have $2^{\aleph_0} = \aleph_2$ and for $([0,1],\text{Borel},\text{Lebesgue})$ to have a splitting. Specifically, he showed that this holds whenever one adds precisely $\aleph_2$ Cohen reals to a model of $\mathsf{CH}$.)

Open question: Does Martin's Axiom (or $\mathsf{PFA}$, or $\mathsf{MM}$) imply that $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting?

Open question: What of the same question in other well-known models of set theory (the random model, Sacks model, Laver model, etc.)?

After chasing down references relating to the paper of Shelah mentioned by Will Brian, I now have a satisfactory answer to the question. It all hinges on whether there is a splitting of the quotient algebra ${\mathcal F}/{\mathcal N}$ of the $\sigma$-algebra ${\mathcal F}$ by the null ideal ${\mathcal N}$, that is to say a Boolean algebra homomorphism $\Phi: {\mathcal F}/{\mathcal N} \to {\mathcal F}$ that is a left inverse for the quotient map $\pi: {\mathcal F} \to {\mathcal F}/{\mathcal N}$.

First suppose that such a map exists. Then for each $\alpha \in A$ and $\omega \in \Omega$ there is a unique element $\tilde X_\alpha(\omega)$ of $\{0,1\}$ with the property that $$ \omega \in \Phi( \pi( X_\alpha^{-1}( \{\tilde X_\alpha(\omega)\} ) ).$$ It is a tedious but routine matter to check that $\tilde X_\alpha: \Omega \to \{0,1\}$ is a modification of $X_\alpha$ (a random variable that agrees almost surely with $X_\alpha$), and that the $\tilde X_\alpha$ satisfy every sentence $S \in {\mathcal S}$ surely (rather than just almost surely).

Conversely, suppose that every family of random variables $X_\alpha$ that almost surely obeys each sentence $S$ in a family ${\mathcal S}$ can be modified to surely obey such a sentence. We consider the family $(X_\alpha)_{\alpha \in {\mathcal F}}$ defined by $$ X_\alpha(\omega) = 1_{\omega \in \alpha}$$ and consider the Boolean algebra homomorphism sentences $$ X_{\alpha \cup \beta} = \max( X_\alpha, X_\beta ); \quad X_{\alpha \cap \beta} = \min(X_\alpha, X_\beta )$$ $$ X_0 = 0; X_1 = 1 $$ $$ X_{\alpha^c} = 1 - X_\alpha$$ for $\alpha, \beta \in {\mathcal F}$, together with the sentences $$ X_\alpha = X_\beta$$ whenever $\alpha,\beta$ differ by a null element in ${\mathcal N}$. Then the indicated random variables $X_\alpha$ obey each these sentences almost surely. By hypothesis, there exists a modification $\tilde X_\alpha$ of each $X_\alpha$ that obey these sentences surely. If we then define $\tilde \Phi: {\mathcal F} \to {\mathcal F}$ by the formula $$ \tilde \Phi(\alpha) := \{ \omega \in \Omega: \tilde X_\alpha(\omega) = 1 \}$$ then one can verify that $\tilde \Phi$ is a Boolean algebra homomorphism such that $\tilde \Phi(\alpha)=\tilde \Phi(\beta)$ whenever $\alpha,\beta$ differ by a null element, and such that $\tilde \Phi(\alpha)$ differs from $\alpha$ by a null element. Thus $\tilde \Phi$ descends to a splitting of ${\mathcal F}/{\mathcal N}$.

As mentioned by Will Brian, the main result of

Shelah, Saharon, Lifting problem of the measure algebra, Isr. J. Math. 45, 90-96 (1983). ZBL0549.03041.

is that it is consistent with ZFC that $[0,1]$ with the Borel sigma-algebra has no splitting; on the other hand it is a classical result of von Neumann and Stone that assuming CH, this measurable space has a splitting. So for this space at least the problem I asked is undecidable in ZFC! On the other hand, the main result in

Maharam, Dorothy, On a theorem of von Neumann, Proc. Am. Math. Soc. 9, 987-994 (1959). ZBL0102.04103.

shows that a splitting always exists for complete probability spaces.