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Diffstat (limited to 'pages/measure-and-integration/lebesgue-integral/convergence-theorems.md')
-rw-r--r-- | pages/measure-and-integration/lebesgue-integral/convergence-theorems.md | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md index 67f0996..f9ebc4a 100644 --- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md +++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md @@ -32,10 +32,10 @@ $$ In the following proof we omit $X$ and $d\mu$ for visual clarity. {% proof %} -By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$, where $g_n = \inf_{k \ge n} f_k$. +By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$, +where $g_n = \inf_{k \ge n} f_k$. Now $(g_n)$ is a monotonic sequence of nonnegative measurable functions. -By the -[Monotone Convergence Theorem](#monotone-convergence-theorem) +By the [Monotone Convergence Theorem](#monotone-convergence-theorem) $$ \int \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int g_n. |