Fatou-Lebesgue theorem

Let $(X,\\mu)$ be a measure space. If $\\Phi\\colon X\\to\\mathbb{R}$ is a nonnegative function with $\\int\\Phi d\\mu<\\infty$ , and if $f_{1},f_{2},\\dots$ is a sequence of measurable functions such that $|f_{n}|\\leq\\Phi$ for each $n$ , then

g=\\liminf_{n\\rightarrow\\infty}f_{n}\\;\\;\\textnormal{and}\\;h=\\limsup_{n%\\rightarrow\\infty}f_{n}

are both integrable, and

-\\infty<\\int gd\\mu\\leq\\liminf_{n\\rightarrow\\infty}\\int f_{n}d\\mu\\leq\\limsup_{k%\\rightarrow\\infty}\\int f_{n}d\\mu\\leq\\int hd\\mu<\\infty.

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