From 4f701090482c5f121ef4118056860f3ba2a10ca8 Mon Sep 17 00:00:00 2001 From: Ildar Gainanov Date: Sat, 17 Jun 2023 16:07:05 +0300 Subject: [PATCH 1/3] fix typos --- mathematical-analysis/sem4/holykpk#blessRNG4.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/mathematical-analysis/sem4/holykpk#blessRNG4.tex b/mathematical-analysis/sem4/holykpk#blessRNG4.tex index c93091c..a91994b 100644 --- a/mathematical-analysis/sem4/holykpk#blessRNG4.tex +++ b/mathematical-analysis/sem4/holykpk#blessRNG4.tex @@ -144,7 +144,7 @@ \subsubsection{Пространство $L^p(E,\mu)$} $L_p := \mathfrak{L}_p /_{\approx}$ --- точки этого пространства. Важно, что эти точки --- классы эквивалентности, как бы не просто функции. Однако, в большинстве случаев удобнее всего работать с каким-то одним представителем класса, ведь, если что, они отличаются только на множестве меры 0. -$[f] = \{g: f \equiv g\}$ +$[f] = \{g: f \approx g\}$ $[f_1] + [f_2] = [f_1 + f_2]$ И введём норму $||[f]|| = \left(\int_{E} |f|^{p}\right)^{\frac{1}{p}}$ @@ -227,7 +227,7 @@ \subsubsection{Образ меры при отображении} \item Пусть $\Phi$ --- ``измеримо'' $\left(\Phi^{-1}(\mathfrak{B}) \subset \mathfrak{A}\right)$ \end{enumerate} -Для $E \in \mathfrak{B}$ зададим $\nu R := \mu\left(\Phi^{-1}(E)\right) = \int_{\Phi^{-1}(E)} 1 d \mu$ +Для $E \in \mathfrak{B}$ зададим $\nu E := \mu\left(\Phi^{-1}(E)\right) = \int_{\Phi^{-1}(E)} 1 d \mu$ $\nu$ --- образ меры $\mu$ при отображении $\Phi$ @@ -1086,7 +1086,7 @@ \subsubsection{Теорема Тонелли} \item $\int_{X \times Y} f dm = \sum c_i \int_{X \times Y} \chi_{C_i} dm=$ вот тут просто раскрываем по пункту для ``ступеньки'' и заносим сумму внутрь $=\sum c_i \int_{X} \left(\int_{Y} \chi_{C_i} d\nu\right) d\mu = \int_{X} \sum c_i \left( \int_{Y} \chi_{C_i} d\nu \right) d\mu = \int_{X} \int_{Y} \left( \sum c_i \chi_{C_i}\right) d\nu d\mu$ \end{enumerate} -\textbf{3. $f \ge 0$ --- ступенчатая} +\textbf{3. $f \ge 0$ --- измеримая} Идея: аппроксимация + теорема Леви $\times \, 2$. \textit{(в этом разделе постоянно используется такой приём --- прим. авт.)} @@ -1333,7 +1333,7 @@ \subsubsection{Критерий плотности} Тогда эквивалентно: \begin{enumerate} - \item $\omega$ --- плотность $\mu$ отностительно $\mu$ + \item $\omega$ --- плотность $\nu$ отностительно $\mu$ \item $\forall A \in \mathfrak{A} \quad \inf_{A} \omega \cdot \mu A \le \nu A \le \sup_{A} \omega \cdot \mu A$ \end{enumerate} @@ -1596,7 +1596,7 @@ \subsubsection{Теорема об интегрировании по парам \item $f: (a, b) \times Y \rightarrow \rinf$ --- ``допустима'', т. е. $f$ суммируема по мере $\lambda_1 \times \mu$ для $\forall t \in (a, b)$ на $(a, t) \times Y$ \item $\mu Y < +\infty$ \item $J(y) = \int_a^{\rightarrow b} f(x, y) dx$ --- равномерно сходится на $Y$ - \item $I(y) := \int_{Y} J(y) d\mu$ + \item $I(y) := \int_{Y} f(y) d\mu$ \end{itemize} Тогда: @@ -3900,7 +3900,7 @@ \subsubsection{Теорема о свойствах аппроксимативн \[|(f * K_h) - f|_1 \goesto{h \rightarrow h_0} \frac{1}{2}(f(x_0 + 0) - f(x_0 - 0))\] \end{enumerate} -\subsubsection{Теорема Фейера } +\subsubsection{Теорема Фейера} \textit{Формулировка:} Аналог теоремы о свойствах АЕ для сумм Фейера From 1abd5fb8c9a390f1128c6fb78ecfd6353f4dc6a5 Mon Sep 17 00:00:00 2001 From: Ildar Gainanov Date: Wed, 6 Sep 2023 00:15:15 +0300 Subject: [PATCH 2/3] Merge --- mathematical-analysis/sem4/holykpk#blessRNG4.tex | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/mathematical-analysis/sem4/holykpk#blessRNG4.tex b/mathematical-analysis/sem4/holykpk#blessRNG4.tex index cee97df..d505556 100644 --- a/mathematical-analysis/sem4/holykpk#blessRNG4.tex +++ b/mathematical-analysis/sem4/holykpk#blessRNG4.tex @@ -142,6 +142,7 @@ \subsubsection{Пространство $L^p(E,\mu)$} $L_p := \mathfrak{L}_p /_{\approx}$ --- точки этого пространства. Важно, что эти точки --- классы эквивалентности, как бы не просто функции. Однако, в большинстве случаев удобнее всего работать с каким-то одним представителем класса, ведь, если что, они отличаются только на множестве меры 0. $[f] = \{g: f \approx g\}$ + $[f_1] + [f_2] = [f_1 + f_2]$ И введём норму $||[f]|| = \left(\int_{E} |f|^{p}\right)^{\frac{1}{p}}$ @@ -240,7 +241,7 @@ \subsubsection{Плотность одной меры по отношению к $X = Y, \mathfrak{A} = \mathfrak{B}, \Phi = $ id -$\nu b = \int_{B} \omega d \mu$ --- ещё одна мера в $X$ +$\nu B = \int_{B} \omega d \mu$ --- ещё одна мера в $X$ Здесь $\omega$ называется плотностью меры $\nu$ относительно меры $\mu$. И в этом случае: @@ -428,7 +429,7 @@ \subsubsection{Теорема Лебега о мажорированной сх \textit{Доказательство:} -Заведём последовательность $h_n := \sup (|f_n - f|, |f_{n + 1} - f|, |f_{n + 2} - f|, \ldots)$. Она убывает, так как по условию у нас есть сходимость почти везде. Также, можно ограничить её: $0 \le h_n \le 2 g$ (модули больше нуля и по условию все $|f_n| \ge g$). А ещё это просто определение последовательности из верхнего предела: +Заведём последовательность $h_n := \sup (|f_n - f|, |f_{n + 1} - f|, |f_{n + 2} - f|, \ldots)$. Она убывает, так как по условию у нас есть сходимость почти везде. Также, можно ограничить её: $0 \le h_n \le 2 g$ (модули больше нуля и по условию все $|f_n| \le g$). А ещё это просто определение последовательности из верхнего предела: \[\lim_{n \rightarrow \infty} h_n = \overline{\lim_{n \rightarrow \infty}} |f_n - f| = 0 \text{ (почти везде)}\] @@ -478,7 +479,7 @@ \subsubsection{Теорема Лебега о мажорированной сх \[\forall \varepsilon > 0 \dbl \exists A \subset X \text{ измеримое } \mu A < + \infty \quad \int_{X \setminus A} g < \varepsilon\] -Если по-русски, то существует некоторое множество в исходном, на котором в основном концентрируется интеграл, следовательно, на остальном кусочке интеграл крайне мал. И мы можем предъявить такое для сколь угодно малого $\varepsilon$. +Если по-русски, то существует некоторое конечное множество в исходном, на котором в основном концентрируется интеграл, следовательно, на остальном кусочке интеграл крайне мал. И мы можем предъявить такое для сколь угодно малого $\varepsilon$. Рассмотрим интеграл как супремум ступенчатых функций: @@ -516,7 +517,7 @@ \subsubsection{Принцип Кавальери} \begin{enumerate} \item при почти всех $x \quad C_{x} \in \mathfrak{B}$ - \item $x \mapsto \nu C_{X}$ --- измеримо на $X$ (сама функция задана почти везде) + \item $x \mapsto \nu C_{x}$ --- измеримо на $X$ (сама функция задана почти везде) \item $m C = \int_{X} \nu (C_{x}) d\mu(x)$ \end{enumerate} @@ -1816,7 +1817,7 @@ \subsubsection{Полнота $L^p$} \[\ldots\] \[\exists n_m: \forall n_k > n_m \dbl ||f_{n_m} - f_{n_k}||_p < \frac{1}{2^m}\] -Соорудим функциональный ряд $S(x) := \sum_{k = 1}^{\infty} |f_{n_{k + 1}} - f_{n_k}|, S(x) \in [0, +\infty]$ (тут уже просто модуль, не перепутайте) и посмотрим на его частичные суммы $S_n$. Его норма ограничена (по неравенству треугольника): +Соорудим функциональный ряд $S(x) := \sum_{k = 1}^{\infty} |f_{n_{k + 1}} - f_{n_k}|, S(x) \in [0, +\infty]$ (тут уже просто модуль, не перепутайте) и посмотрим на его частичные суммы $S_n$. Их норма ограничена (по неравенству треугольника): \[||S_n||_p \le \sum_{k = 1}^{n} ||f_{n_{k + 1}} - f_{n_k}||_p \le 1\] From 73cf3ff945928214941f7557b0f746fb790177c7 Mon Sep 17 00:00:00 2001 From: Ildar Gainanov Date: Thu, 21 Sep 2023 23:29:39 +0300 Subject: [PATCH 3/3] Fix typos --- .../sem4/holykpk#blessRNG4.pdf | Bin 1227712 -> 1227690 bytes .../sem4/holykpk#blessRNG4.tex | 16 ++++++++-------- 2 files changed, 8 insertions(+), 8 deletions(-) diff --git a/mathematical-analysis/sem4/holykpk#blessRNG4.pdf b/mathematical-analysis/sem4/holykpk#blessRNG4.pdf index 0dcac033521e2bda273524e4a45e10a7e1045f89..48df0a1e137e2a7090e8371de67465b8f8c0ddb3 100644 GIT binary patch delta 42036 zcmV)MK)Aoa_e!ewO0bJY0XLWN+y*Is?OIE39LEv9^H)r6T$t1DM^$wu%R5X3(})z9gk?b)5BNLc}L5SH5Qsds(#=#kU=Nlw4NlKJzucUND3 zYy6}@7$7~}Jx&gV#KTh}8>M$!M<@JM6_Q3!?7?KYL!#_G0#oq&e9A-nd zoCv>|bP|a_?I9crf1=`t`5)>`Kpo!Q{o(6xQBFz;j~IoJ)2zL)&S(B2N}!R6NS)|I zAjb4<If`$$Vk~q!HsRxO2z|aV1 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\Var{a}{y} f + f(y) - \Var{a}{x} - f(x) = \Var{x}{y} f + (f(y) - f(x)) \ge 0\]