# Question concerning Fubini theorem

Let $f:(a,b)\times(a,b)\times (a,b)\rightarrow \mathbb R$ be integrable. Then by Fubini theorem for a.e $(x,y) \in (a,b)\times(a,b)$ there exists the integral $\int_a^b f(x,y,z)dz$.

Is is then true that for almost each $(x,y)\in (a,b)\times(a,b)$ the integral

$\int_0^h \int_a^b f(x+t,y,z)dt dz$

exist, where $0<h<b-x$?

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