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 for an introduction and this answer 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.

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.

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 for an introduction and this answer 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.

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.