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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-29 17:32:24 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-29 17:32:24 +0100 |
commit | a1b5de688d879069b5e1192057d71572c7bc5368 (patch) | |
tree | a0f4801d14bfbcc75a6091bdc7d17aceab71f6d4 /pages/measure-and-integration/lebesgue-integral | |
parent | 8b9bb9346c217874670b0f1798ab6f1cb28fdb83 (diff) | |
download | site-a1b5de688d879069b5e1192057d71572c7bc5368.tar.zst |
Update
Diffstat (limited to 'pages/measure-and-integration/lebesgue-integral')
-rw-r--r-- | pages/measure-and-integration/lebesgue-integral/convergence-theorems.md | 6 | ||||
-rw-r--r-- | pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md | 22 |
2 files changed, 25 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 1a34820..6808280 100644 --- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md +++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md @@ -4,9 +4,9 @@ parent: Lebesgue Integral grand_parent: Measure and Integration nav_order: 2 description: > -We state and prove the most important convergence theorems of Lebesgue -integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the -Dominated Convergence Theorem. + We state and prove the most important convergence theorems of Lebesgue + integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the + Dominated Convergence Theorem. --- # {{ page.title }} 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 8482e87..0424117 100644 --- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md +++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md @@ -23,6 +23,28 @@ Endowed with pointwise addition and scalar multiplication $\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space. {% endproposition %} +{% proof %} +We show that $\mathscr{L}^p := \mathscr{L}^p(X,\mathcal{A},\mu)$ is a linear subspace of +the vector space of all $\KK$-valued functions on $X$. +The set $\mathscr{L}^p$ is nonempty since +it contains the zero function. +Now, suppose $f$ and $g$ are in $\mathscr{L}^p$. +Then the sum $f+g$ is measurable, because $f$ and $g$ are measurable. +Moreover, the function $\abs{f+g}^p$ is integrable, because we have the estimate + +$$ +\abs{f+g}^p +\le (\abs{f} + \abs{g})^p +\le \big\lparen 2 \max(\abs{f},\abs{g}) \big\rparen^p +\le 2^p (\abs{f}^p + \abs{g}^p), +$$ + +where $\abs{f}^p$ and $\abs{g}^p$ are integrable. +This proves that $f+g$ lies in $\mathscr{L}^p$. +Finally, it is easy to see that $\alpha f$ lies in $\mathscr{L}^p$ +for any scalar $\alpha \in \KK$. +{% endproof %} + {% proposition %} $\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$. {% endproposition %} |