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Suppose I have a Bochner integrable function, $t\mapsto u(t)\in X$, with $X$ a separable Banach space, and $0\leq t\leq T<\infty$. If I introduce a mapping $f:[0,T]\times X\to X$, under what assumptions will $t\mapsto f(t,u(t))$ be Bochner integrable? Is joint measurabiity of $f$ sufficient? Continuity?

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