The disintegration theorem says that under certain conditions, a probability measure $\mu$ on a measurable space the existence of
Let $Y$ and $X$ be two Radon spaces (i.e. separable metric spaces on which every probability measure is a Radon measure). Let $μ ∈ P(Y)$, let $π : Y → X$ be a Borel-measurable function, and let $ν ∈ P(X)$ be the pushforward measure from $Y$ to $X$ by $π$. Then there exists a $ν$-almost everywhere uniquely determined family of probability measures $\{μ_x\}_{x∈X} ⊆ P(Y)$ such that
- the function $x \mapsto \mu_{x}$ is Borel measurable, in the sense that $x \mapsto \mu_{x} (B)$ is a Borel-measurable function for each
Borel-measurable set $B ⊆ Y$;- $μ_x$ lives on the fiber $π^{-1}(x)$: for $ν$-almost all $x ∈ X$, \mu_{x} \left( Y \setminus \pi^{-1} (x) \right) = 0, and so $\mu_x(E) = \mu_x(E \cap \pi^{-1}(x));$
- for every Borel-measurable function $f : Y → [0, +∞]$, \int_{Y} f(y) \, \mathrm{d} \mu (y) = \int_{X} \int_{\pi^{-1} (x)} f(y) \, \mathrm{d} \mu_{x} (y) \mathrm{d} \nu (x).
- I was wondering if the probability measures can be relaxed to measures in the disintegration theorem?
when $Y = X_1 × X_2$ and $π_i : Y → X_i$ is the natural projection, we can apply the disintegration theorem, and get the result
each fibre $π_1^{-1}(x1)$ can be canonically identified with $X_2$ and there exists a Borel family of probability measures $\{ \mu_{x_{1}} \}_{x_{1} \in X_{1}}$ in $P(X_2)$ (which is $(π_1)∗(μ)$-almost everywhere uniquely determined) such that \mu = \int_{X_{1}} \mu_{x_{1}} \, \mu \left(\pi_1^{-1}(\mathrm d x_1) \right)= \int_{X_{1}} \mu_{x_{1}} \, \mathrm{d} (\pi_{1})_{*} (\mu) (x_{1}), $
I wonder if this result is still true if Y$, $X_i$ are not required to be Radon spaces but just general measure spaces as long as $Y = X_1 × X_2$ and $π_i : Y → X_i is the natural projection?
In other words, given two measurable spaces X_1$ and $X_2$ and a measure on the product measurable space $X_1 \times X_2$ , what are some necessary and/or sufficient conditions for the measure on $X_1 \times X_2$ to be the composition of some measure on $X_1$ and some transition measure from $X_1$ to $X_2$?
Thanks and regards!