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diff --git a/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md new file mode 100644 index 0000000..a77cf9a --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md @@ -0,0 +1,27 @@ +--- +title: Almost Everywhere +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 1 +--- + +# {{ page.title }} + +{% definition Almost Everywhere %} +We say that a property $P(x)$ depending on $x \in X$ +holds *almost everywhere* (abbreviated by *a.e.*) or for *almost all $x \in X$* if +the set of points where it does not hold has measure zero. +{% enddefinition %} + +In other words, $P(x)$ a.e. iff +$\mu(\set{x \in X : \neg P(x)}) = 0$. + +{% theorem %} +Let $f : X \to \overline{\RR}$ be a nonnegative measurable function. Then + +$$ +\int_X f \, d\mu = 0 +$$ + +holds if and only if $f$ vanishes almost everywhere. +{% endtheorem %} diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md new file mode 100644 index 0000000..67f0996 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md @@ -0,0 +1,77 @@ +--- +title: Convergence Theorems +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 2 +--- + +# {{ page.title }} + +For all statements on this page, +assume that $(X,\mathcal{A},\mu)$ is a measure space. + +{% theorem * Monotone Convergence Theorem %} +For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ be a measurable function. +If $0 \le f_n \le f_{n+1}$ almost everywhere, then + +$$ +\int_X \lim_{n \to \infty} f_n \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu. +$$ +{% endtheorem %} + +Note that the pointwise limit $\lim_{n \to \infty} f_n$ always exists and is measurable by this proposition. + +{% lemma * Fatou’s Lemma %} +For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ be a nonnegative measurable function. Then + +$$ +\int_X \liminf_{n \to \infty} f_n \, d\mu \le \liminf_{n \to \infty} \int_X f_n \, d\mu. +$$ +{% endlemma %} + +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 $g_n = \inf_{k \ge n} f_k$. +Now $(g_n)$ is a monotonic sequence of nonnegative measurable functions. +By the +[Monotone Convergence Theorem](#monotone-convergence-theorem) + +$$ +\int \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int g_n. +$$ + +For all $k \ge n$ one has $g_n \le f_k$, hence +$\int g_n \le \int f_k$ by the monotonicity of the integral. +This implies + +$$ +\int g_n \le \inf_{k \ge n} \int f_k +$$ + +for all $n \in \NN$. In the limit $n \to \infty$ we obtain + +$$ +\lim_{n \to \infty} \int g_n +\le \liminf_{n \to \infty} \int f_n +$$ + +thereby completing the proof. +{% endproof %} + +{% theorem * Dominated Convergence Theorem %} +Let $(X,\mathcal{A},\mu)$ be a measure space. +For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ (or $\CC$) be a measurable function. +Suppose that the pointwise limit $f = \lim_{n \to \infty} f_n$ exists almost everywhere. +Suppose further that there exists an integrable function $g : X \to \overline{\RR}$ +such that $\abs{f_n} \le g$ almost everywhere for all $n \in \NN$. +Then the functions $f_n$ and $f$ are all integrable, and + +$$ +\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu. +$$ +{% endtheorem %} + +{% proof %} +TODO +{% endproof %} diff --git a/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md new file mode 100644 index 0000000..6e5179c --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md @@ -0,0 +1,14 @@ +--- +title: Fubini Theorem +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 2 +--- + +# {{ page.title }} + +{% theorem %} +{% endtheorem %} + +{% proof %} +{% endproof %} diff --git a/pages/measure-and-integration/lebesgue-integral/index.md b/pages/measure-and-integration/lebesgue-integral/index.md new file mode 100644 index 0000000..a857d95 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/index.md @@ -0,0 +1,112 @@ +--- +title: Lebesgue Integral +parent: Measure and Integration +nav_order: 2 +has_children: true +has_toc: false +--- + +# {{ page.title }} + +For this entire section we fix a measure space $(X,\mathcal{A},\mu)$. + +## Integration of Nonnegative Step Functions + +{% definition %} +Let $f : X \to \RR$ be a nonnegative step function +with representation $f = \sum_{i=1}^n \alpha_i \chi_{A_i}$, +where $\alpha_1, \ldots, \alpha_n \ge 0$ and +$A_1, \ldots, A_n \in \mathcal{A}$. +We define the *integral of $f$ on $X$ with respect to $\mu$* by + +$$ +\int_X f \, d\mu = \sum_{i=1}^n \alpha_i \, \mu(A_i) \in [0,\infty]. +$$ +{% enddefinition %} + +TODO: This does not depend on the representation of $f$. + +## Integration of Nonnegative Measurable Functions + +{% theorem Approximation by Step Functions %} +Every nonnegative measurable function $f : X \to \overline{\RR}$ +is the pointwise limit of an increasing sequence $(s_n)$ of +nonnegative step functions $s_n : X \to \RR$. +{% endtheorem %} + +{% definition %} +Let $f : X \to \overline{\RR}$ be a nonnegative measurable function +and let $(s_n)$ be a sequence of nonnegative step functions +with $s_n \uparrow f$. +We define the *integral of $f$ on $X$ with respect to $\mu$* by + +$$ +\int_X f \, d\mu = \lim_{n \to \infty} \int_X s_n \, d\mu \in [0,\infty]. +$$ +{% enddefinition %} + +## Integrable Functions + +Recall that the positive and (flipped) negative parts +of a function $f : X \to \overline{R}$ are defined by + +$$ +f^+ = \max(f,0) \qquad +f^- = \max(-f,0), +$$ + +and that $f$ is measurable if and only if both $f^+$ and $f^-$ are measurable. +We have $f = f^+\! - f^-$. + +{% definition Integrable Function, Lebesgue Integral %} +A measurable function $f : X \to \overline{\RR}$ is said to be +*integrable on $X$ with respect to $\mu$* if the integrals + +$$ +\int_X f^+ \, d\mu, \qquad \int_X f^- \, d\mu +\tag{$*$} +$$ + +are both finite. +In this case the *(Lebesgue) integral of $f$ on $X$ with respect to $\mu$* is defined as + +$$ +\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \RR. +$$ +{% enddefinition %} + +Sometimes it is convenient to have a slightly more general notion of integrability: + +{% definition Quasi-Integrable Function %} +A measurable function $f : X \to \overline{\RR}$ is said to be +*quasi-integrable on $X$ with respect to $\mu$* if at least one of the integrals +$(*)$ is finite. +In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as + +$$ +\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \overline{\RR}. +$$ +{% enddefinition %} + +{% definition %} +A measurable function $f : X \to \CC$ is said to be +*integrable on $X$ with respect to $\mu$* if +$\Re f$ and $\Im f$ are integrable on $X$ with respect to $\mu$. +In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as + +$$ +\int_X f \, d\mu = \int_X \Re f \, d\mu + i \int_X \Im f \, d\mu \in \CC. +$$ +{% enddefinition %} + +## Integration on Measurable Subsets + +{% definition %} +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 + +$$ +\int_A f \, d\mu = +\int_X \chi_A f \, d\mu. +$$ +{% 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 new file mode 100644 index 0000000..023c253 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md @@ -0,0 +1,36 @@ +--- +title: The L<sup>p</sup> Spaces +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 4 +--- + +# {{ page.title }} + +{% definition %} +Let $(X,\mathcal{A},\mu)$ be a measure space and let $p \in [1,\infty)$. +We write $\mathscr{L}^p(X,\mathcal{A},\mu)$ for the set of all +measurable functions $f : X \to \KK$ such that $\abs{f}^p$ is integrable. +For such $f$ we write + +$$ +\norm{f}_p = {\bigg\lparen\int_X \abs{f}^p \, d\mu\bigg\rparen}^{\!1/p}. +$$ +{% enddefinition %} + +{% proposition %} +Endowed with pointwise addition and scalar multiplication +$\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space. +{% endproposition %} + +{% proposition %} +$\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$. +{% endproposition %} + +{% theorem * Young Inequality %} +Consider $p,q > 1$ such that $1/p + 1/q = 1$. Then + +$$ +a \cdot b \le \frac{a^p}{p} + \frac{b^q}{q} \qquad \forall a,b \ge 0. +$$ +{% endtheorem %} diff --git a/pages/measure-and-integration/lebesgue-integral/transformation-formula.md b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md new file mode 100644 index 0000000..6f02bc8 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md @@ -0,0 +1,14 @@ +--- +title: Transformation Formula +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 3 +--- + +# {{ page.title }} + +{% theorem %} +{% endtheorem %} + +{% proof %} +{% endproof %} |