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Let $f$ and $g$ be non-negative, integrable functions on a measure space with measure $\mu$, and suppose there is some constant $c > 0$ such that for every $t \geq 0$, the inequality $\mu(\{f \geq t\}) \geq c ( \{g \geq t\} )$ holds. How can we show that $||f||_1 \geq ||g||_1$ ?

I assume that we need to use some type of convergence theorem, but I am not sure how to set up the convergence. For example, I know that $ \int g \ 1_{ \{g \leq n\}} \to \int g$ as $n \to \infty$, but I'm not seeing how to use the given inequality.

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