Yes, there is such a category, namely the category of measurable spaces and morphisms of measurable spaces equipped with fiberwise measures (also known as operator valued weights).
See [this answer](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820) for an introduction and [this answer](https://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th/49542#49542) for a list of further references.

Intuitively, a morphism f: X→Y in this category is a morphism of measurable spaces
together with a measure on each fiber of f that varies measurably with respect to Y.
In particular, if Y is the point, then a morphism f: X→pt is simply a measurable space equipped with a measure,
i.e., a _measured_ space.

As David Roberts has already pointed out, the categorical construction that captures the intuition of the question is called the [dependent sum](http://ncatlab.org/nlab/show/dependent+product).

In the notation of the nLab artcle we have B=Z, A=X, I=pt,
the morphism f: A→I is the measure a, the morphism g: B→A is the projection map Z=X×Y→X equipped with the family of measures b_x.

The dependent sum of g indexed by f exists and is the composition fg,
which in our case is the measure c.