Uniqueness theorem for probability measures?

Question

I could swear that years ago I saw a statement of a result saying that any nonatomic (complete?) probability measure was in some sense isomorphic to Lebesgue measure on $[0,1]$. Wanted to try to prove it just to keep in shape, but I can't come up with a precise statement of such a result that's not obviously false.

Anyone recall what that result actually says?

Context: Was reminded of this by the comments to my answer here .

My work so far: Realized at the start that it can't be a pointwise isomorphism, because there could be a null set of cardinality greater than $c$.

So then I say maybe it's the measure algebras that are isomorphic (where the measure algebra for a measure space is the algebra of measurable sets modulo null sets). But no, the measure algebra for the uniform measure on $\{0,1\}^S$ can have arbitrarily large cardinality (quick proof: There's a family of independent events of cardinality $|S|$.)

Currently I'm saying maybe it's a result for nonatomic Borel probability measures on a compact metric space. That eliminates the two counterexamples above; the closest I've got to a proof is verifying that another thing I seemed to recall is actually true: Any compact metric space is the continuous image of the Cantor set.

Answer 1

Example 3.6.2 in Bogachev's Measure Theory (Volume 1) says that

If $(\Omega, \mathcal A,P)$ is a probability space, and $X:\Omega\to \mathbb R$ is a random variable such that its cdf $F_X$ is continuous, then the pushforward of $P$ by $F_X\circ X$ is the Lebesgue measure on $[0,1]$.

To prove this, it suffices to check that for all $t\in [0,1]$, $P_{F_X\circ X}([0,t])=t$.
Indeed $P_{F_X\circ X}([0,t])=P_X(\{x, F_X(x)\leq t\})=P_X(\{x, x\leq F_X^{-1}(t)\})=F_X(F_X^{-1}(t)) = t$
where $F_X^{-1}(t)=\sup\{x, F_X(t)\leq t\}$.

Chapter $9$ of the second volume is entirely dedicated to transformations of measures and isomorphisms.