Copied from Real Analysis (Stein) .
To make it easier to remember what the author said previously while learning.
目录
- 1 Measure Theory
- 2 Integration Theory
- 1 The Lebesgue integral: basic properties and convergence theorems
- 2 The space \(L^1\) of integrable functions
- 3 Fubini's theorem
1 Measure Theory
1 Preliminaries
Open, closed, and compact sets
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Rectangles and cubes
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Lemma 1.1
If a rectangle is the almost disjoint union of finitely many other rectangles, say \(R=\bigcup_{k=1}^N R_k\), then
\[|R|=\sum_{k=1}^N |R_k| \]Lemma 1.2
If \(R,R_1,\dots,R_N\) are rectangles, and \(R\subset \bigcup_{k=1}^N R_k\), then
\[|R|\le \sum_{k=1}^N |R_k| \]Theorem 1.3
Every open subset \(\mathcal O\) of \(\mathbb R\) can be written uniquely as a countable union of disjoint open intervals.
Theorem 1.4
Every open subset \(\mathcal O\) of \(\mathbb R^d,d\ge 1\), can be written as a countable union of almost disjoint closed cubes.
The Cantor set
2 The exterior measure
Definition
If \(E\) is any subset of \(\mathbb R^d\), the exterior measure of \(E\) is
\[m_*(E)=\inf \sum_{j=1}^\infty |Q_j| \]where the infimum is taken over all countable coverings \(E\subset \bigcup_{j=1}^\infty Q_j\) by closed cubes.
Properties of the exterior measure
Observation 1 (Monotonicity)
If \(E_1\subset E_2\), then \(m_*(E_1)\le m_*(E_2)\).
Observation 2 (Countable sub-additivity)
If \(E=\bigcup_{j=1}^\infty E_j\), then \(m_*(E)\le \sum_{j=1}^\infty m_*(E_j)\)
Observation 3 (Presented by open sets)
If \(E\subset \mathbb R^d\), then \(m_*(E)=\inf m_*(\mathcal O)\), where the infimum is taken over all open sets \(\mathcal O\) containing E.
Observation 4 (Additivity if disjoint)
If \(E=E_1\bigcup E_2\), and \(d(E_1,E_2)>0\), then
\[m_*(E)=m_*(E_1)+m_*(E_2) \]Observation 5 (Additivity for countable almost disjoint cubes)
If a set \(E\) is the countable union of almost disjoint cubes \(E=\bigcup_{j=1}^\infty Q_j\), then
\[m_*(E)=\sum_{j=1}^\infty|Q_j| \]3 Measurable sets and the Lebesgue measure
Definition
A subset \(E\) of \(\mathbb R^d\) is Lebesgue measurable or simply measurable, if for any \(\epsilon>0\) there exists an open set \(\mathcal O\) with \(E\subset \mathcal O\) and
\[m_*(\mathcal O -E)\le \epsilon \]Property 1
Every open set in $\mathbb R^d $ is measurable.
Property 2
If \(m_*(E)=0\), then \(E\) is measurable. In particular, if \(F\) is a subset of a set of exterior measure \(0\), then \(F\) is measurable.
Property 3
A countable union of measurable sets is measurable.
Property 4
Closed sets are measurable.
Lemma 3.1
If \(F\) is closed, \(K\) is compact, and these sets are disjoint, then \(d(F,K)>0\)
Property 5
The complement of a measurable set is measurable.
Property 6
A countable intersection of measurable sets is measurable.
Theorem 3.2
If \(E_1,E_2,\dots,\) are disjoint measurable sets, and \(E=\bigcup_{j=1}^\infty E_j\), then
\[m(E)=\sum_{j=1}^\infty m(E_j) \]Corollary 3.3
Suppose \(E_1,E_2,\dots\) are measurable subsets of \(\R^d\).
- If \(E_k\nearrow E\), then \(m(E)=\lim_{N\to \infty} m(E_N)\).
- If \(E_k\searrow E\) and \(m(E_k)<\infty\) for some \(k\), then \(m(E)=\lim_{N\to \infty} m(E_N)\)
Theorem 3.4
Suppose \(E\) is a measurable subset of \(\mathbb R^d\). Then, for every \(\epsilon>0\):
- There exists an open set \(\mathcal O\) with \(E\subset \mathcal O\) and \(m(\mathcal O-E)\le \epsilon\).
- There exists a closed set \(F\) with \(F\subset E\) and \(m(E-F)\le \epsilon\).
- If \(m(E)\) is finite, there exists a compact set K with \(K\subset E\) and \(m(E-K)\le \epsilon\).
- If \(m(E)\) is finite, there exists a finite union \(F=\bigcup_{j=1}^N Q_j\) of closed cubes such that \(m(E\triangle F)\le \epsilon\)
Invariance properties of Lebesgue measure
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\(\sigma\)-algebras and Borel sets
From the point of view of the Borel sets, the Lebesgue sets arise as the completion of the \(\sigma\)-algebra of Borel sets, that is ,by adjoining all subsets of Borel sets of measure zero.
Corollary 3.5
A subset \(E\) of \(\mathbb R^d\) is measurable
- if and only if \(E\) differs from a \(G_\delta\) by a set of measure zero.
- if and only if \(E\) differs from an \(F_\sigma\) by a set of measure zero.
Construction of a non-measurable set
4 Measurable functions
simple function and step function
A simple function is a finite sum
\[f=\sum_{k=1}^N a_k \chi_{E_k} \]where each \(E_k\) is a measurable set of finite measure, and the \(a_k\) are constants.
4.1 Definition and basic properties
Property 1
The finite-valued function \(f\) is measurable if and only if \(f^{-1}(\mathcal O)\) is measurable for every open set \(\mathcal O\), and if and only if \(f^{-1}(F)\) is measurable for every closed set \(F\).
Property 2
If \(f\) is continuous on \(\mathbb R^d\), then \(f\) is measurable.
If \(f\) is measurable and finite-valued, and \(\Phi\) is continuous, then \(\Phi\circ f\) is measurable.
Property 3
Suppose \(\{f_n\}_{n=1}^\infty\) is a sequence of measurable functions. Then
\[\sup_n f_n(x),\ \inf_n f_n(x),\ ,\limsup_n f_n(x),\ \liminf_n f_n(x),\ \]are measurable.
Property 4
If \(\{f_n\}_{n=1}^\infty\) is a collection of measurable functions, and \(\lim_{n\to\infty} f_n(x)=f(x)\), then \(f\) is measurable.
Property 5
If \(f\) and \(g\) are measurable, then
- The integer powers \(f^k,k\ge 1\) are measurable.
- \(f+g\) and \(fg\) are measurable if both \(f\) and \(g\) are finite-valued.
Property 6
Suppose \(f\) is measurable, and \(f(x)=g(x)\) for a.e. x. Then \(g\) is measurable.
4.2 Approximation by simple functions or step functions
Theorem 4.1
Suppose \(f\) is a non-negative measurable function on \(\mathbb R^d\). Then there exists an increasing sequence of non-negative simple functions \(\{\varphi_k\}_{k=1}^\infty\) that converges pointwise to \(f\), namely,
\[\varphi_k(x)\le \varphi_{k+1}(x)\ and \ \lim_{k\to \infty} \varphi_k(x)=f(x),\ \forall x \]Theorem 4.2
Suppose \(f\) is measurable on \(\mathbb R^d\). Then there exists a sequence of simple functions \(\{\varphi_k\}_{k=1}^\infty\) that satisfies
\[|\varphi_k(x)|\le |\varphi_{k+1}(x)|\ and \ \lim_{k\to \infty} \varphi_k(x)=f(x),\ \forall x \]Theorem 4.3
Suppose \(f\) is measurable on \(\mathbb R^d\). Then there exists a sequence of step functions \(\{\psi_k\}_{k=1}^\infty\) that converges pointwise to \(f(x)\) for almost every \(x\).
4.3 Littlewood's three principles
Description
To provide a useful intuitive guide in the initial study of the theory.
- Every set is nearly a finite union of intervals.
- Every function is nearly continuous.
- Every convergent sequence is nearly uniformly convergent.
Theorem 4.4 (Egorov)
Suppose \(\{f_k\}_{k=1}^\infty\) is a sequence of measurable functions defined on a measurable set \(E\) with \(m(E)<\infty\), and assume that \(f_k\to f\) a.e. on E. Given \(\epsilon>0\), we can find a closed set \(A_\epsilon \subset E\) such that \(m(E-A_\epsilon)\le \epsilon\) and \(f_k\to f\) uniformly on \(A_\epsilon\).
Theorem 4.5 (Lusin)
Suppose \(f\) is measurable and finite valued on \(E\) with \(E\) of finite measure. Then for every \(\epsilon>0\) there exists a closed set \(F_\epsilon\) with
\[F_\epsilon\subset E,\ and \ m(E-F_\epsilon)\le \epsilon \]and such that \(f|_{F_\epsilon}\) is continuous.
5 The Brunn-Minkowski inequality
Theorem 5.1
Suppose \(A\) and \(B\) are measurable sets in \(\mathbb R^d\) and their sum \(A+B\) is also measurable. Then the inequality
\[m(A+B)^{1/d}\ge m(A)^{1/d}+m(B)^{1/d} \]holds.
2 Integration Theory
1 The Lebesgue integral: basic properties and convergence theorems
Stage one: simple functions
\[\varphi(x)=\sum_{k=1}^N a_k \chi_{E_k}(x) \]The canonical form of \(\varphi\) is the unique decomposition as above, where the numbers \(a_k\) are distinct and non-zero, and the sets \(E_k\) are disjoint.
If \(\varphi\) is a simple function with canonical form \(\varphi(x)=\sum_{k=1}^N a_k \chi_{F_k}(x)\), then we define the Lebesgue integral of \(\varphi\) by
\[\int_{\mathbb R^d} \varphi(x)dx=\sum_{k=1}^M c_km(F_k) \]If \(E\) is a measurable subset of \(\mathbb R^d\) with finite measure, then \(\varphi(x)\chi_E(x)\) is also a simple function, and we define
\[\int_E \varphi(x)dx=\int \varphi(x)\chi_E(x) dx \]Proposition 1.1
The integral of simple functions defined above satisfies the following properties:
- Independence of the representation.
- Linearity.
- Additivity.
- Monotonicity.
- Triangle inequality.
Stage two: bounded functions supported on a set of finite measure
The support of a measurable function \(f\) is defined to be the set of all points where \(f\) does not vanish,
\[supp(f)=\{x|f(x)\neq 0\} \]We shall also say that \(f\) is supported on a set \(E\), if \(f(x)=0\) whenever \(x\notin E\).
Lemma 1.2
Let \(f\) be a bounded function supported on a set \(E\) of finite measure. If \(\{\varphi_n\}_{n=1}^\infty\) is any sequence of simple functions bounded by \(M\), supported on \(E\), and with \(\varphi_n(x)\to f(x)\) for a.e. \(x\), then:
- The limit \(\lim_{n\to \infty} \int \varphi_n\) exists.
- If \(f=0\) a.e., then the limit \(\lim_{n\to \infty} \int \varphi_n\) equals \(0\).
Definition
We can now turn to the integration of bounded functions that are supported on sets of finite measure. For such a function \(f\) we define its Lebesgue integral by
\[\int f(x)dx=\lim_{n\to \infty}\int \varphi_n(x) dx \]where \(\{\varphi_n\}\) is any sequence of simple functions satisfying: \(|\varphi_n|\le M\), each \(\varphi_n\) is supported on the support of \(f\), and \(\varphi_n(x)\to f(x)\) for a.e. \(x\), as \(n\) tends to infinity.
Proposition 1.3
Suppose \(f\) and \(g\) are bounded functions supported on sets of finite measure. Then the following properties hold.
- Linearity.
- Additivity.
- Monotonicity.
- Triangle inequality.
Theorem 1.4 (Bounded convergence theorem)
Suppose that \(\{f_n\}\) is a sequence of measurable functions that are all bounded by \(M\), are supported on a set \(E\) of finite measure, and \(f_n(x)\to f(x)\) a.e. \(x\) as \(n\to \infty\). Then \(f\) is measurable, bounded, supported on \(E\) for a.e. \(x\), and
\[\int|f_n-f|\to 0\ \ \ as\ n\to \infty \]Consequently,
\[\int f_n\to \int f\ \ \ as \ n\to \infty \]Theorem 1.5
Suppose \(f\) is Riemann integrable on the closed interval \([a,b]\). Then \(f\) is measurable, and
\[\int_{[a,b]}^{\mathcal R} f(x)dx=\int_{[a,b]}^{\mathcal L} f(x)dx \]where the integral on the left-hand side is the standard Riemann integral and that on the right-hand side is the Lebesgue integral.
Stage three: non-negative functions
We proceed with the integrals of functions that are measurable and non-negative but not necessarily bounded. It will be important to allow these functions to be extended-valued, that is, these functions may take on the value \(+\infty\) (on a measurable set).
In the case of such a function \(f\) we define its (extended) Lebesgue integral by
\[\int f(x) dx=\sup_g \int g(x)dx \]where the supremum is taken over all measurable functions \(g\) such that \(0\le g \le f\), and where \(g\) is bounded and supported on a set of finite measure.
Proposition 1.6
The integral of non-negative measurable functions enjoys the following properties:
- Linearity.
- Additivity.
- Monotonicity.
- If \(g\) is integrable and \(0\le f\le g\), then \(f\) is integrable.
- If \(f\) is integrable, then \(f(x)<\infty\) for almost every \(x\).
- If \(\int f=0\), then \(f(x)=0\) for almost every \(x\).
Lemma 1.7 (Fatou)
Suppose \(\{f_n\}\) is a sequence of measurable functions with \(f_n\ge 0\). If \(\lim_{n\to \infty} f_n(x)=f(x)\) for a.e. \(x\), then
\[\int f\le \liminf_{n\to \infty} \int f_n \]Corollary 1.8
Suppose \(\{f\}\) is a non-negative measurable function, and \(\{f_n\}\) a sequence of non-negative measurable functions with \(f_n(x)\le f(x)\) and \(f_n(x)\to f(x)\) for almost every \(x\). Then
\[\lim_{n\to \infty} \int f_n=\int f \]Corollary 1.9 (Monotone convergence theorem)
Suppose \(\{f_n\}\) is a sequence of non-negative measurable functions with \(f_n\nearrow f\). Then
\[\lim_{n\to \infty}\int f_n=\int f \]Corollary 1.10
Consider a series \(\sum_{k=1}^\infty a_k(x)\), where \(a_k(x)\ge 0\) is measurable for every \(k\ge 1\). Then
\[\int \sum_{k=1}^\infty a_k(x) dx=\sum_{k=1}^\infty\int a_k(x) dx \]If \(\sum_{k=1}^\infty \int a_k(x) dx\) is finite, then the series \(\sum_{k=1}^\infty a_k(x)\) converges for a.e. \(x\).
Stage four: general case
If \(f\) is any real-valued measurable function on \(\mathbb R^d\), we say that \(f\) is Lebesgue integrable (or just integrable) if the non-negative measurable function \(|f|\) is integrable in the sense of the previous section.
If \(f\) is Lebesgue integrable. We define the Lebesgue integral of \(f\) by
\[\int f=\int f^+-\int f^- \]where \(f^+(x)=\max(f(x),0),f^-(x)=\max(-f(x),0)\).
Proposition 1.11
The integral of Lebesgue integrable functions is linear, additive, monotonic, and satisfies the triangle inequality.
Proposition 1.12
Suppose \(f\) is integrable on \(\mathbb R^d\). Then for every \(\epsilon>0\),
-
There exists a set of finite measure \(B\) (a ball, for example) such that
\[\int_{B^c}|f|<\epsilon \] -
There is a \(\delta>0\) such that
\[\int_E |f|<\epsilon\ \ whenever \ m(E)<\delta \]
The last condition is known as absolute continuity.
Theorem 1.13 (Dominated convergence theorem)
Suppose \(\{f_n\}\) is a sequence of measurable functions such that \(f_n(x)\to f(x)\) a.e. \(x\), as \(n\) tends to infinity. If \(|f_n(x)|\le g(x)\), where \(g\) is integrable, then
\[\int |f_n-f|\to 0 \ \ as \ n\to \infty \]and consequently
\[\int f_n\to \int f \ \ as \ n\to \infty \]Complex-valued functions
We say that \(f=u+iv\) is Lebesgue integrable if the function \(|f(x)|=(u(x)^2+v(x)^2)^{1/2}\) is Lebesgue integrable.
A complex-valued function is integrable if and only if both its real and imaginary parts are integrable. Then, the Lebesgue integral of \(f\) is defined by
\[\int f(x)dx=\int u(x) dx+i\int v(x) dx \]2 The space \(L^1\) of integrable functions
Definition: Norm
For any integrable function \(f\) on \(\mathbb R^d\) we define the norm of \(f\)
\[||f||=||f||_{L^1}=||f||_{L^1(\mathbb R^d)}=\int_{\mathbb R^d}|f(x)| dx \]Proposition 2.1
Suppose \(f\) and \(g\) are two functions in \(L^1(\mathbb R^d)\).
- \(||af||_{L^1(\mathbb R^d)}=|a|||f||_{L^1(\mathbb R^d)},\forall a\in \mathbb C\)
- \(||f+g||_{L^1(\mathbb R^d)}\le ||f||_{L^1(\mathbb R^d)}+||g||_{L^1(\mathbb R^d)}\)
- \(||f||_{L^1(\mathbb R^d)}=0\) if and only if \(f=0\) a.e.
- \(d(f,g)=||f-g||_{L^1(\mathbb R^d)}\) defines a metric on \(L^1(\mathbb R^d)\).
Definition: Complete
A space \(V\) with a metric \(d\) is said to be complete if for every Cauchy sequence \(\{x_k\}\) in \(V\) (that is, \(d(x_k,x_l)\to 0\) as \(k,l\to \infty\)) there exists \(x\in V\) such that \(\lim_{k\to\infty} x_k=x\) in the sense that
\[d(x_k,x)\to 0,\ \ as \ k \to \infty \]Theorem 2.2 (Riesz-Fischer)
The vector space \(L^1\) is complete in its metric.
Corollary 2.3
If \(\{f_n\}_{n=1}^\infty\) converges to \(f\) in \(L^1\), then there exists a subsequence \(\{f_{n_k}\}_{k=1}^\infty\) such that
\[f_{n_k}(x)\to f(x) \ \ a.e. \ x. \]Definition: Dense
A family \(\mathcal G\) of integrable functions is dense in \(L^1\) if for any \(f\in L^1\) and \(\epsilon>0\), there exists \(g\in \mathcal G\) so that \(||f-g||_{L^1}<\epsilon\).
Theorem 2.4
The following families of functions are dense in \(L^1(\mathbb R^d)\):
- The simple functions
- The step functions.
- The continuous functions of compact support.
Extension
If \(E\) is a subset of positive measure, we can define \(L^1(E)\) and carry out the arguments that are analogous to \(L^1(\mathbb R^d)\).
Invariance Properties
translation-invariance:
\[\int_{\mathbb R^d}f(x-h)dx=\int_{\mathbb R^d} f(x)dx \]dilation-invariance:
\[\delta^d\int_{\mathbb R^d} f(\delta x)dx=\int_{\mathbb R^d} f(x) dx \]reflection-invariance:
\[\int_{\mathbb R^d} f(-x) dx=\int_{\mathbb R^d} f(x)dx \]Translations and continuity
Proposition 2.5
Suppose \(f\in L^1(\mathbb R^d)\). Then
\[||f_h-f||_{L^1}\to 0 \ \ as \ h\to 0 \]3 Fubini's theorem
Definition: slice
If \(f\) is a function in \(\mathbb R^{d_1}\times \mathbb R^{d_2}\), the slice of \(f\) corresponding to \(y\in \mathbb R^{d_2}\) is the function \(f^y\) of the \(x\in \mathbb R^{d_1}\) variable, given by \(f^y(x)=f(x,y)\). Similarly, the slice of \(f\) for a fixed \(x\in \mathbb R^{d_1}\) is \(f_x(y)=f(x,y)\).
In the case of a set \(E\in \mathbb R^{d_1}\times\mathbb R^{d_2}\), we define its slice by
\[E^y=\{x\in \mathbb R^{d_1}|(x,y)\in E\}\ \ \ E^x=\{y\in\mathbb R^{d_2}|(x,y)\in E\} \]3.1 Statement and proof of the theorem
Suppose \(f(x,y)\) is integrable on \(\mathbb R^{d_1} \times \mathbb R^{d_2}\). Then for almost every \(y\in \mathbb R^{d_2}\):
-
The slice \(f^y\) is integrable on \(\mathbb R^{d_1}\).
-
The function defined by \(\int_{\mathbb R^{d_1}} f^y(x)dx\) is integrable on \(\mathbb R^{d_2}\).
-
Moreover:
\[\int_{\mathbb R^{d_2}}\left(\int_{\mathbb R^{d_1}}f(x,y)dx\right)dy=\int_{\mathbb R^d} f \]
proof
-
(Outline) Let \(\mathcal F\) denote the set of integrable functions on \(\mathbb R^d\) which satisfy all three conclusions in the theorem and set out to prove that \(L^1(\mathbb R^d)\subset \mathcal F\).
-
First, show that \(\mathcal F\) is closed under operations such as linear combinations (Step 1) and limits (Step 2).
-
Then, begin to construct families of functions in \(\mathcal F\)
-
Since any integrable function is the "limit" of simple functions, and simple functions are themselves linear combinations of sets of finite measure,
-
the goal quickly becomes to prove that \(\chi_E\) belongs to \(\mathcal F\) whenever \(E\) is a measurable subset of \(\mathbb R^d\) with finite measure.
-
To achieve this goal, we begin with rectangles and work our way up to sets of type \(G_\delta\) (Step 3), and sets of measure zero (Step 4).
-
-
Finally, a limiting argument shows that all integrable functions are in \(\mathcal F\).
-
-
(closed under linear combinations) Any finite linear combination of functions in \(\mathcal F\) also belongs to \(\mathcal F\).
-
(closed under limits) Suppose \(\{f_k\}\) is a sequence of measurable functions in \(\mathcal F\) so that \(f_k\nearrow f\) or \(f_k\searrow f\), where \(f\) is integrable (on \(\mathbb R^d\)). Then \(f\in \mathcal F\).
-
Any characteristic function of a set \(E\) that is a \(G_\delta\) and of finite measure belongs to \(\mathcal F\).
- Suppose \(E\) is a bounded open cube.
- Suppose \(E\) is a subset of boundary of some closed cube.
- Suppose \(E\) is a finite union of closed cubes whose interiors are disjoint.
- Prove that if \(E\) is open and of finite measure, then \(\chi_E\in \mathcal F\).
- In general, if \(E\) is a \(G_\delta\)of finite measure, then \(\chi_E\in \mathcal F\).
-
If \(E\) has measure \(0\), then \(\chi_E\) belongs to \(\mathcal F\).
-
If \(E\) is any measurable subset of \(\mathbb R^d\) with finite measure, then \(\chi_E\) belongs to \(\mathcal F\).
-
If \(f\) is integrable, then \(f\in \mathcal F\).