Monotone Convergence Theorem

Monotone Convergence Theorem

Theorem

Suppose $\{f_{n}\}$ is a sequence of non-negative measurable functions, $f_{n}:X\to [0,+\infty ]$ such that $f_{n-1}\leq f_{n}$ for all $n$. Furthermore, $\lim _{n\to +\infty }f_{n}=f(=\sup _{n}f_{n})$. Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to+\infty} \int f_n = \int \lim_{n\to+\infty} f_n .<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.2 </ref>}