In this section, \((\Omega, \mathcal{F}, \mu)\) is taken to be a measure space. Also, unless otherwise stated, any simple function \(f=\sum_{i=1}^m a_i I_{A_i}\) is assumed to be \(\mathcal{F}\)-measurable and to be represented in such a way that \(A_i\in \mathcal{F}\) and \(A_i=\phi\) for all \(i=1,\ldots,m\), and \(\Omega=\uplus_{i=1}^m A_i\).
If \(f=\sum_{i=1}^m a_iI_{A_i}\) and \(g=\sum_{j=1}^n b_jI_{B_j}\) are measurable, extended-real-valued simple functions satisfying \(0\le f(\omega) \le g(\omega)\) for all \(\omega\in \Omega\), then \[ \sum_{i=1}^m a_i\mu(A_i)\le\sum_{j=1}^n b_j\mu(B_j). \]
매우매우 중요
Assume in this definition that \(f:\Omega\rightarrow\bar{\mathbb{R}}\) is measurable-\(\mathcal{F}/\mathcal{R}\).
If \(f=\sum_{i=1}^m a_iI_{A_i}\) is a nonnegative, real-valued simple function, then the integral of \(f\) (with respect to \(\mu\)) is defined to be \[ \int f \mbox{ }d\mu=\sum_{i=1}^m a_i\mu(A_i). \]
For general nonnegative \(f\), \[ \int f\mbox{ } d\mu=\sup\left\{ \int g\mbox{ }d\mu :0\le g\le f, \mbox{ } g \mbox{ simple, real-valued, and measurable.} \right\} \]
Finally, for general \(f\), we define \[ \int f\mbox{ } d\mu=\int f^+\mbox{ } d\mu-\int f^-\mbox{ } d\mu \] whenever the difference on the right-hand side is well-defined. In this case, we say that the integral exists, and otherwise that the integral fails to exist.
If both \(\int f^+\mbox{ } d\mu\) and \(\int f^-\mbox{ } d\mu\) are finite, so that \(\int f\mbox{ } d\mu\) exists and is finite, then we say that \(f\) is integrable w.r.t \(\mu\) \((\int |f|d\mu <\infty)\).
Simple function \(f\)에 대해서 integral은 finite partition들에 대한 measure들의 sum으로 계산된다.
Nonnegative general \(f\)의 integral은 f보다 작거나 같은 simple function의 integral값들의 supremum값이다.
General \(f\)의 integral은 \(f^+\ge 0\)와 \(f^-\ge 0\)값으로 나누고, 각각을 Nonnegative general \(f\)의 integral로 구하여 계산한다. 이 때 이 integral값이 finite하다면 \(f\)는 integrable하다.
If \(f\ge 0\) is measurable-\(\mathcal{F}/\mathcal{R}\) and \(f_n, n\ge 1\) is a sequence of real-valued simple functions satisfying \(0\le f_n \uparrow f\), then \[ 0\le \int f_n \mbox{ }d\mu \uparrow \int f\mbox{ }d\mu. \]
A property is said to hold almost everywhere(a.e.), or more specifically \(\mu\)-almost everywhere (\(\mu\)-a.e.), if the property holds for all \(\omega\) in some set \(A\in \mathcal{F}\) with \(\mu(A^c)=0\). If \(\mu\) is a probability measure, then we say that the property holds almost surely (a.s.) or with probability 1 (w.p.1).
어떤 property가 \(\mu(A)=1\), \(\mu(A^c)=0\)인 some set \(A\in\mathcal{F}\)에서 만족한다면 이 property는 almost everywhere 하게 hold한다고 말할 수 있다.
Measure값이 0인 partition of \(\Omega\)에 대해서는 고려하지 않는 의미로 생각할 수 있다.
If \(f,g:\Omega\rightarrow \bar{\mathbb{R}}\) are measurable functions satisfying \(f=g\) a.e., then \(\int f\mbox{ }d\mu\) exists if and only if \(\int g\mbox{ }d\mu\) exists, and in this case \(\int f\mbox{ }d\mu= \int g\mbox{ }d\mu\). In particular, if \(f=0\) a.e., then \(\int f\mbox{ }d\mu= 0\).
포인트는 만약 \(f=g\) a.e. 이고 \(f,g\)의 integral들이 exist한다면(not \(\infty-\infty\)), \(\int f\mbox{ }d\mu= \int g\mbox{ }d\mu\)한다는 것이다.
증명은 \(f=g\) a.e.를 이용한다. \(A\in \mathcal{F}\)가 \(f(\omega)=g(\omega)\) for all \(\omega\in A\), where \(\mu(A^c)=0\)을 만족하는 set이라 하면 \(\int f\mbox{ }d\mu= \int fI_A \mbox{ }d\mu= \int gI_A \mbox{ }d\mu=\int g\mbox{ }d\mu\)이다 (Exercise).
Suppose that \(f,g:\Omega\rightarrow\bar{\mathbb{R}}\) are measurable-\(\mathcal{F}\). If \(0\le f\le g\) a.e., then \(\int f\mbox{ }d\mu \le \int g\mbox{ }d\mu\).
\(0\le f\le g\) a.e.라 하자 (\(0\le fI_A \le gI_A\) for \(A\in \mathcal{F}\) with \(\mu(A^c)=0\)).
\(\int f\mbox{ }d\mu= \int fI_A \mbox{ }d\mu\)이고 \(\int gI_A \mbox{ }d\mu=\int g\mbox{ }d\mu\)이다.
때문에 WLOG, \(0\le f(\omega)\le g(\omega)\) for all \(\omega\in \Omega\)라고 가정하자.
Simple function \(h\)에 대해 \(\int f \mbox{ }d\mu:=\sup\left\{ \int h\mbox{ }d\mu: 0\le h \le f, \mbox{ }\mbox{ h real valued, simple, measurable} \right\}\)이다. 즉 \(\int f\mbox{ }d\mu\)는 \(\int h\mbox{ }d\mu\) 의 upper bound이다. 하지만 이는 \(\int g\mbox{ }d\mu\)라는 더 우측의 upper bound에 bounded되어있다(End).
매우 중요
Suppose that \(f\) and \(f_n,n\ge 1\) are \(\mathcal{F}\)-measurable, extended-real-valued functions on \(\Omega\).
If \(0\le f_n\uparrow f\) a.e., then \[ \int f_n \mbox{ }d\mu \uparrow \int f \mbox{ }d\mu. \]
Suppose throughout that \(f,g:\Omega\rightarrow \bar{\mathbb{R}}\) are measurable-\(\mathcal{F}\).
(Additivity) If \(f+g\) is well-defined(not \(\infty-\infty\)) and \(\int f\mbox{ }d\mu\) and \(\int g\mbox{ }d\mu\) exist, then \[ \int (f+g)\mbox{ }d\mu= \int f\mbox{ }d\mu+ \int g\mbox{ }d\mu, \]
as long as the expression on the right-hand side is defined. In particular, if \(f\) and \(g\) are integrable, then so is \(f+g\).
(Linearity) Suppose that \(a,b\in \mathbb{R}\), that \(af+bg\) is well-defined, and that \(\int f\mbox{ }d\mu\) and \(\int g\mbox{ }d\mu\) exist. Then, \[ \int (af+bg)\mbox{ }d\mu= a\int f\mbox{ }d\mu+ b\int g\mbox{ }d\mu, \]
as long as the expression on the right-hand side is defined. In particular, this holds whenever \(f\) and \(g\) are integrable.
If \(f\ge 0\) and \(\mu(f>0)>0\), then \(\int f\mbox{ }d\mu>0\).
\(\mu(f>0)>0\)은 \(A:=\{w\in \Omega: f(\omega)>0\}\)가 공집합이 아니라는 의미이다.
\(A^c=\{w\in \Omega: f(\omega)=0\}\)이므로 \(\mu(A^c)=\int_{A^c} f\mbox{ }d\mu=\int fI_{A^c}\mbox{ }d\mu=0\)이다. 때문에 \(f>0\) a.e.이다.
if \(f=0\) a.e., then \(\int f\mbox{ }d\mu= 0\)의 역명제로 if \(f>0\) a.e., then \(\int f\mbox{ }d\mu>0\)을 유추할 수 있다(두가지 경우밖에 없기 때문).
If \(\int f\mbox{ }d\mu<\infty\), then \(f<\infty\) a.e.
(Monotonicity) If \(f\le g\) a.e., and both \(\int f\mbox{ }d\mu\) and \(\int g\mbox{ }d\mu\) exist, then \(\int f\mbox{ }d\mu\le \int g\mbox{ }d\mu\).
If \(\int f\mbox{ }d\mu\) exists, then \(|\int f\mbox{ }d\mu|\le \int |f|\mbox{ }d\mu\).
Assume that \(f\le g\) a.e. If \(\int f\mbox{ }d\mu\) exists and \(\int f\mbox{ }d\mu>-\infty\), then \(\int g\mbox{ }d\mu\) exists.
Similarly, if \(\int g\mbox{ }d\mu\) exists and \(\int g\mbox{ }d\mu<\infty\), then \(\int f\mbox{ }d\mu\) exists. In either case, \(\int f\mbox{ }d\mu\le \int g\mbox{ }d\mu\).