Let $f : X\rightarrow Y$ be a measurable function between measure spaces $X,Y$ with measures denoted $\mu,\nu$ respectively. Suppose singleton subsets of $Y$ are measurable; hence fibers of $f$ are measurable subsets in $X$.
My question is, in general does there exist a collection $\{\lambda_y\}_{y\in Y}$ where $\lambda_y$ is a measure on $f^{-1}(y)$ for each $y\in Y$, such that for every measurable subset $A \subset X$ we have $$ \mu(A) = \int_{y \in Y} \big(\lambda_y(A \cap f^{-1}(y))\big)\,d\nu(y) \;\;? $$
If not in general, then does this hold when, for example, $\mu,\nu$ are $\sigma$-finite and there is some kind of absolute continuity condition, as in the Radon-Nikodym theorem?
My guess would be that this is somehow related to the R-N theorem; however, I'm not sure how to construct any precise relationship between the two situations. Would anyone have any suggestions or hints on how to think about this?
The answer is yes, this decomposition exists if the measure spaces $X$ and $Y$ are Lebesgue spaces and if $f$ sends $\mu$ to $\nu$. The fibers of $f$ define a measurable partition of $X$ on which we can decompose the measure $\mu$. This is called the disintegration of the measure with respect to the partition and goes back to a famous article of Rokhlin in the 40s.
The disintegration theorem is used in ergodic theory, for example, to define the ergodic decomposition of a probability measure invariant by some transformation of $X$: the measure can be written as an integral of ergodic measures defined on a partition of the space into invariant sets, known as the ergodic components of the system.