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### Bochner integral: Is $f=g$ $\mu$-a.e. if their integrals are equal on every measurable set?
I fill some detail in @geetha290krm's excellent answer to better understand his/her ideas. We have  \varphi \left (\int_A f \mathrm d \mu\right ) = \varphi \left (\int_A g \mathrm d \mu \right) \...