Let $(X_1,\mu_1)$ and $(X_2,\mu_2)$ be two $\sigma$-finite measure spaces. Let $f(x_1,x_2)$ be a measurable complex valued function. Stein and Shakarchi's 4th book asks to prove that $$ \|{\int f(x_1,x_2)d\mu_2}\|_{L^p(X_1)}\leq\int \|f(x_1,x_2)\|_{L^p(X_1)}d\mu_2 $$ when the right side of the inequality is finite. I don't understand why we need the right side to be finite. Even if the right side were $+\infty$, won't the inequality still remain trivially true as the left side can be $+\infty$ at max? I think that the right side being finite has something to do with $\int f(x_1,x_2)d\mu_2$ being well defined. How can I show that this is true?

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    $\begingroup$ Edit: Applying the usual Minkowski inequality for real valued functions to the function $|f(x_1,x_2)|$ and using the fact that the right side is finite helped in concluding that $\int f(x_1,x_2)d\mu_2$ is well defined. $\endgroup$ – Manan Jun 18 '17 at 9:05

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