$\textbf{Beppo Levi's theorem for Lebesgue integral}$
Let $(X,\mathscr{A},\mu)$ be a measure space and let $\sum_{i=1}^\infty f_i$ be an infinite series of $[0,+\infty]$ valued $\mathscr{A}$-Borel-measurable functions on $X$. Then $$\int\sum_{i=1}^\infty f_id\mu=\sum_{i=1}^\infty\int f_id\mu$$
$\textbf{Beppo Levi's theorem for Lebesgue integral}$
Let $(X,\mathscr{A},\mu)$ be a measure space and let $\sum_{i=1}^\infty f_i$ be an infinite series of $[0,+\infty]$ valued $\mathscr{A}$-Borel-measurable functions on $X$. Then $$\int\sum_{i=1}^\infty f_id\mu=\sum_{i=1}^\infty\int f_id\mu$$
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๐๐๐ฉ๐ฉ๐จย ๐๐๐ฏ๐ขโ๐ฌย ๐ญ๐ก๐๐จ๐ซ๐๐ฆย ๐๐จ๐ซย ๐๐๐๐๐ฌ๐ ๐ฎ๐ย ๐ข๐ง๐ญ๐๐ ๐ซ๐๐ฅ
Let (๐,๐,๐) be a measure space and let โแตขโโโแชฒ ๐แตข be an infinite series of [0,+โ] valued ๐-Borel-measurable functions on ๐. Then
โซโแตขโโโแชฒ ๐แตข๐๐=โแตขโโโแชฒ โซ๐แตข๐๐