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Let's say I have the measure spaces $(X,B,\mu), (Y,C,\nu)$. $m$ is the standard product measure. Can something interesting be said about a measurable $f(x,y)$ given I know $h_f(x) = \int_{Y} f(x,y) d\nu$? i.e., what is required of $f$ having certain integrals on each "slice"?

It seems that $h$ partitions the measurable $f$'s into equivalence classes by $f\sim g$ iff $\int_{Y} (f-g)(x,y) d\nu(y) = 0$ - hence it is closely related to solving $h_f = 0$. Are there interesting solutions to this, except for functions of $y$? What about $h_f=1$?

The "reverse" problem seems simple, so maybe I am missing something: $h_f$ can be any $B$-measurable function, by taking $f$ to be that function (i.e., only a function of $x$).

It's sort of an open question, I am not sure where it leads but I feel that answers can sharpen my understanding of product measures.

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