diff options
Diffstat (limited to 'pages/measure-and-integration/lebesgue-integral')
3 files changed, 5 insertions, 5 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. diff --git a/pages/measure-and-integration/lebesgue-integral/index.md b/pages/measure-and-integration/lebesgue-integral/index.md index a857d95..3418e10 100644 --- a/pages/measure-and-integration/lebesgue-integral/index.md +++ b/pages/measure-and-integration/lebesgue-integral/index.md @@ -106,7 +106,7 @@ For any measurable subset $A \subset X$ we define the *integral on $A$* of a (quasi-)integrable function $f : X \to \overline{\RR}$ (or $\CC$) by $$ -\int_A f \, d\mu = +\int_A f \, d\mu = \int_X \chi_A f \, d\mu. $$ {% enddefinition %} diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md index 023c253..8482e87 100644 --- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md +++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md @@ -1,5 +1,5 @@ --- -title: The L<sup>p</sup> Spaces +title: The L<sup>p</sup> Spaces parent: Lebesgue Integral grand_parent: Measure and Integration nav_order: 4 |