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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-15 05:11:07 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-15 05:11:07 +0100 |
commit | 7c66b227a494748e2a546fb85317accd00aebe53 (patch) | |
tree | 9c649667d2d024b90b32d36ca327ac4b2e7caeb2 /pages/measure-and-integration/lebesgue-integral/convergence-theorems.md | |
parent | 28407333ffceca9b99fae721c30e8ae146a863da (diff) | |
download | site-7c66b227a494748e2a546fb85317accd00aebe53.tar.zst |
Update
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. |