Construction of random elements in Hilbert space which are almost surely orthogonal.

Counterexample. This counterexample is inspired by class notes by D.J.H. Garling from the 1980s.

Let $\kappa$ be a measurable cardinal https://en.wikipedia.org/wiki/Measurable_cardinal which is also a limit ordinal. Thus there is a $\{0,1\}$ valued measure on $\kappa$ in which every subset is measurable.

Let $\mathcal H$ be the Hilbert space whose basis is of cardinality $\kappa$, and let $(e_{\alpha})_{\alpha \in \kappa}$ be a basis. Let $X,Y:\kappa \to \mathcal H$ be the functions $X(\alpha) = e_{\alpha}$ and $Y(\alpha) = e_{\alpha'}$, where $\alpha'$ denotes the successor ordinal of $\alpha$.

Clearly $\langle X,Y\rangle = 0$ everywhere. It remains to show that $X$ and $Y$ are independent. But this follows because any two subsets of $\kappa$ are independent (since their measures can only take the values $0$ or $1$).

It does use measurable cardinals, whose existence cannot be proved. But most likely, if their existence can be disproved, then probably the same proof will show ZF is inconsistent.