For any non-negative real-valued Lebesgue integrable function $ f $ on $ \mathbb{R}^n $, does the inequality $$ \left( \int_{\mathbb{R}^n} f \: \mathrm{d}x \right)^p \leq \int_{\mathbb{R}^n} f^p \: \mathrm{d}x . $$ hold true?
2026-05-15 20:56:00.1778878560
Integral raised to power p \leq integral where integrand is raised to power p.
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This is false, as can be seen by testing against indicator functions $\chi_A$ of measurable sets.