Let $(X,\mathcal{X},\mu)$ be a standard probability space. A measure $\mu$ is atomic if $\mu$ is supported on at most countable many atoms, and is atomless if $\mu$ has no atom (a point x is an atom if $\mu(\{x\})>0$). 


Now let $(X,\mathcal{X},\mu)$ be an extension of $(Y,\mathcal{Y},\nu)$ and $\mu=\int\_{Y}\mu\_{y}d\nu(y)$ be the disintegration of $\mu$ with respect of $\nu$. Then for each $\mu\_{y}$, there is a unique way to write $\mu\_{y}=\mu\_{y,c}+\mu\_{y,d}$, where $\mu\_{y,c}$ is an atomless and $\mu\_{y,d}$ is an atomic measure on $(X,\mathcal{X})$.


Question: let $\mu\_{c}=\int\_{Y}\mu\_{y,c}d\nu(y)$ and $\mu\_{d}=\int\_{Y}\mu\_{y,d}d\nu(y)$. Is the decomposition $\mu=\mu\_{c}+\mu\_{d}$ well defined? i.e. Is $A$ measurable for $\mu\_{c}$ and $\mu\_{d}$ for all $A\in\mathcal{X}$?



decomposing a measure into relative atom and atomless parts
It is generally true that if $(X,\mathcal{X})$ is a measurable space, $(Y,\mathcal{Y})$ a standard Borel space, and $\kappa:X\to\mathcal{P}(X)$ a transition probability, then the measure-valued function that maps $x$ to the atomic part of $\kappa(x)$ is measurable. The rest is a red herring.


With the usual $\sigma$-algebra on $\mathcal{P}(X)$ (the space of probability measures on $X$) generated by evaluation of measurable sets, the function $j:\mathcal{P}(X)\times X\to [0,1]$ given by $j(\mu,x)=\mu\big(\{x\}\big)$ is measurable. You can find the details in Appendix A of "Fundamentals of Nonparametric Bayesian Inference" (2017) by Ghosal and van der Vaart. Let $C\subseteq\mathcal{P}(X)\times X$ be the set on which $j$ is strictly positive. Let $g\_\mu$ be the $\mu$-section of $C$. By a standard generalization of Fubini's theorem for transition probabilities, the function $\mu\mapsto \int g\_\mu~\mathrm d\mu$ is measurable and so is the function $\mu\mapsto \int\_A g\_\mu~\mathrm d\mu$, which gives you the desired measurability.


For another approach, look at Theorem 2.12 of



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> Dubins, Lester, and David Freedman. "[Measurable sets of measures](https://msp.org/pjm/1964/14-4/pjm-v14-n4-p07-s.pdf)."
>  Pacific Journal of Mathematics 14.4 (1964): 1211-1222.
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