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 $A\in\mathcal{X}$?