Let $M$ be the Lebesgue measure algebra of the unit interval $[0,1]$, i.e. equivalence classes of Lebesgue measurable sets modulo sets of measure $0$. This is a complete Boolean algebra, hence in particular a frame. Let $\Omega(\mathbb{R})$ be the standard topology on $\mathbb{R}$. $\Omega(\mathbb{R})$ is also a frame. A frame homomorphism is a function between frames preserving arbitrary sups and finite meets (in particular, $\bot$ and $\top$).
Any measurable function $f : [0,1] \to \mathbb{R}$ induces a frame homomorphism $\Omega(\mathbb{R}) \to M$ by taking (equivalence classes of) preimages.
My question is, does every frame homomorphism $F : \Omega(\mathbb{R}) \to M$ arise from a measurable map $f : [0,1] \to \mathbb{R}$?