측도론과 확률론 요약 정리
📂確率論 측도론과 확률론 요약 정리 概要 測度論と確率論を学んだ人向けの定義と概念の要約資料です。迅速な復習と定義の参照のために作成されました。
測度論 代数 X ≠ ∅ X \ne \varnothing X = ∅ コレクション A \mathcal{A} A 有限 合集合と補集合に対して閉じている時、これを代数 と言います。
可算 合集合に対して閉じている代数をσ \sigma σ
Note:
定義により A \mathcal{A} A ( ∵ E 1 ∩ E 2 = ( E 1 ∪ E 2 ) c ∈ A \big( \because E_{1} \cap E_{2} = \left( E_{1} \cup E_{2} \right)^{c} \in \mathcal{A} ( ∵ E 1 ∩ E 2 = ( E 1 ∪ E 2 ) c ∈ A E 1 , E 2 ∈ A ) E_{1}, E_{2} \in \mathcal{A} \big) E 1 , E 2 ∈ A ) A \mathcal{A} A ∅ \varnothing ∅ X X X ( ∵ E ∈ A \big( \because E \in \mathcal{A} ( ∵ E ∈ A ⟹ \implies ⟹ ∅ = E ∩ E c ∈ A and X = E ∪ E c ∈ A ) \varnothing = E \cap E^{c} \in \mathcal{A} \text{ and } X = E \cup E^{c} \in \mathcal{A} \big) ∅ = E ∩ E c ∈ A and X = E ∪ E c ∈ A ) X X X X X X σ \sigma σ X X X ボレル σ \sigma σ と言い、B X \mathcal{B}_{X} B X
ボレル σ \sigma σ σ \sigma σ E \mathcal{E} E X X X σ \sigma σ ( X , E ) (X, \mathcal{E}) ( X , E ) 可測空間 と言い、E ∈ E E \in \mathcal{E} E ∈ E 可測集合 と言います。
特に言及がない限り、以下では固定された可測空間 ( X , E ) (X, \mathcal{E}) ( X , E )
可測関数 全ての実数 α ∈ R \alpha \in \mathbb{R} α ∈ R f : X → R f : X \to \mathbb{R} f : X → R E \mathcal{E} E 可測 と言います。
{ x ∈ X : f ( x ) > α } ∈ E ∀ α ∈ R .
\left\{ x \in X : f(x) \gt \alpha \right\} \in \mathcal{E}\qquad \forall \alpha \in \mathbb{R}.
{ x ∈ X : f ( x ) > α } ∈ E ∀ α ∈ R .
一般化 ( X , E ) (X, \mathcal{E}) ( X , E ) ( Y , F ) (Y, \mathcal{F}) ( Y , F ) f : X → Y f : X \to Y f : X → Y ( E , F ) (\mathcal{E}, \mathcal{F}) ( E , F ) 可測 と言います。
f − 1 ( F ) = { x ∈ X : f ( x ) ∈ F } ∈ E ∀ F ∈ F .
f^{-1}(F) = \left\{ x \in X : f(x) \in F \right\} \in \mathcal{E}\qquad \forall F \in \mathcal{F}.
f − 1 ( F ) = { x ∈ X : f ( x ) ∈ F } ∈ E ∀ F ∈ F .
Note: E \mathcal{E} E ( Y , F ) = ( R , B R ) (Y, \mathcal{F}) = (\mathbb{R}, \mathcal{B}_{\mathbb{R}}) ( Y , F ) = ( R , B R )
測度 E \mathcal{E} E ( X , E ) (X, \mathcal{E}) ( X , E ) X X X 測度 とは、次を満たす関数 μ : E → [ 0 , ∞ ] \mu : \mathcal{E} \to [0, \infty] μ : E → [ 0 , ∞ ]
Null empty set: μ ( ∅ ) = 0 \mu (\varnothing) = 0 μ ( ∅ ) = 0 Countable additivity: { E j } \left\{ E_{j} \right\} { E j } E \mathcal{E} E μ ( ⋃ j E j ) = ∑ j μ ( E j ) \displaystyle \mu \left( \bigcup\limits_{j} E_{j} \right) = \sum\limits_{j} \mu (E_{j}) μ ( j ⋃ E j ) = j ∑ μ ( E j ) 三つ組 ( X , E , μ ) (X, \mathcal{E}, \mu) ( X , E , μ ) 測度空間 と言います。特に言及がない限り以下では固定された測度空間 ( X , E , μ ) (X, \mathcal{E}, \mu) ( X , E , μ )
ボレル測度 とは、定義域がボレル σ \sigma σ B R \mathcal{B}_{\mathbb{R}} B R μ : B R → [ 0 , ∞ ]
\mu : \mathcal{B}_{\mathbb{R}} \to [0, \infty]
μ : B R → [ 0 , ∞ ]
( X , E ) (X, \mathcal{E}) ( X , E ) ( Y , F ) (Y, \mathcal{F}) ( Y , F ) μ \mu μ ν \nu ν E × F \mathcal{E} \times \mathcal{F} E × F μ × ν \mu \times \nu μ × ν μ \mu μ ν \nu ν 積測度 と言います。
μ × ν ( E × F ) = μ ( E ) ν ( F ) for all rectangles E × F .
\mu \times \nu (E \times F) = \mu (E) \nu (F)\qquad \text{ for all rectangles } E \times F.
μ × ν ( E × F ) = μ ( E ) ν ( F ) for all rectangles E × F .
積分 実関数 f f f 単純 と言います。
単純可測関数 φ \varphi φ φ = ∑ j = 1 n a j χ E j , where E j = φ − 1 ( { a j } ) and range ( φ ) = { a 1 , … , a n } .
\begin{equation}
\varphi = \sum\limits_{j=1}^{n} a_{j}\chi_{E_{j}}, \text{ where } E_{j} = \varphi^{-1}(\left\{ a_{j} \right\}) \text{ and } \operatorname{range} (\varphi) = \left\{ a_{1}, \dots, a_{n} \right\}.
\end{equation}
φ = j = 1 ∑ n a j χ E j , where E j = φ − 1 ( { a j } ) and range ( φ ) = { a 1 , … , a n } . χ E j \chi_{E_{j}} χ E j E j E_{j} E j φ \varphi φ standard representation と言います。
φ \varphi φ ( 1 ) (1) ( 1 ) μ \mu μ φ \varphi φ ∫ φ d μ : = ∑ j = 1 n a j μ ( E j ) .
\int \varphi d\mu := \sum\limits_{j=1}^{n} a_{j}\mu (E_{j}).
∫ φ d μ := j = 1 ∑ n a j μ ( E j ) . Notation:
∫ φ d μ = ∫ φ = ∫ φ ( x ) d μ ( x ) , ∫ = ∫ X .
\int \varphi d\mu = \int \varphi = \int \varphi(x) d\mu (x), \qquad \int = \int_{X}.
∫ φ d μ = ∫ φ = ∫ φ ( x ) d μ ( x ) , ∫ = ∫ X .
f f f ( X , E ) (X, \mathcal{E}) ( X , E ) μ \mu μ f f f ∫ f d μ : = sup { ∫ φ d μ : 0 ≤ φ ≤ f , φ is simple and measurable } .
\int f d\mu := \sup \left\{ \int \varphi d\mu : 0 \le \varphi \le f, \varphi \text{ is simple and measurable} \right\}.
∫ fd μ := sup { ∫ φ d μ : 0 ≤ φ ≤ f , φ is simple and measurable } .
f : X → R f : X \to \mathbb{R} f : X → R 正の部分 と負の部分 をそれぞれ次のように定義します。
f + ( x ) : = max ( f ( x ) , 0 ) ) , f − 1 ( x ) : = min ( − f ( x ) , 0 ) ) .
f^{+}(x) := \max \left( f(x), 0 \right)),\qquad f^{-1}(x) := \min \left(-f(x), 0 \right)).
f + ( x ) := max ( f ( x ) , 0 ) ) , f − 1 ( x ) := min ( − f ( x ) , 0 ) ) . ∫ f + \displaystyle \int f^{+} ∫ f + ∫ f − \displaystyle \int f^{-} ∫ f − f f f 積分可能 と言います。また∣ f ∣ = f + − f − \left| f \right| = f^{+} - f^{-} ∣ f ∣ = f + − f −
積分可能な実関数たちの集合はベクトル空間 であり、積分はこのベクトル空間上の線形汎関数 です。このベクトル空間を次のように表記します。
L = L ( X , E , μ ) = L ( X , μ ) = L ( X ) = L ( μ ) , L = L 1
L = L(X, \mathcal{E}, \mu) = L(X, \mu) = L(X) = L(\mu), \qquad L = L^{1}
L = L ( X , E , μ ) = L ( X , μ ) = L ( X ) = L ( μ ) , L = L 1
L p L^{p} L p
測度空間( X , E , μ ) (X, \mathcal{E}, \mu) ( X , E , μ ) 0 < p < ∞ 0 \lt p \lt \infty 0 < p < ∞ L p L^{p} L p L p ( X , E , μ ) : = { f : X → R ∣ f is measurable and ( ∫ ∣ f ∣ p d μ ) 1 / p < ∞ } .
L^{p}(X, \mathcal{E}, \mu) := \left\{ f : X \to \mathbb{R} \left| f \text{ is measurable and } \left( \int \left| f \right|^{p} d\mu \right)^{1/p} \lt \infty \right. \right\}.
L p ( X , E , μ ) := { f : X → R f is measurable and ( ∫ ∣ f ∣ p d μ ) 1/ p < ∞ } .
確率論 表記法と用語 Analysts’ Term Probabilists’ Term Measure space ( X , E , μ ) such that μ ( X ) = 1 Probability space ( Ω , F , P ) Measure μ : E → R such that μ ( X ) = 1 Probability P : F → R ( σ -)algebra E on X ( σ -)field F on Ω Mesurable set E ∈ E Event E ∈ F Measurable real-valued function f : X → R Random variable X : Ω → R Integral of f , ∫ f d μ Expextation of f , E ( X ) f is L p X has finite p th moment Almost everywhere, a.e. Almost surely, a.s.
\begin{array}{lll}
\text{Analysts’ Term} && \text{Probabilists’ Term} \\
\hline
\text{Measure space } (X, \mathcal{E}, \mu) \text{ such that } \mu (X) = 1 && \text{Probability space } (\Omega, \mathcal{F}, P) \\
\text{Measure } \mu : \mathcal{E} \to \mathbb{R} \text{ such that } \mu (X) = 1 && \text{Probability } P : \mathcal{F} \to \mathbb{R} \\
(\sigma\text{-)algebra E \mathcal{E} E X X X F \mathcal{F} F Ω \Omega Ω p p p Analysts’ Term Measure space ( X , E , μ ) such that μ ( X ) = 1 Measure μ : E → R such that μ ( X ) = 1 ( σ -)algebra E on X Mesurable set E ∈ E Measurable real-valued function f : X → R Integral of f , ∫ fd μ f is L p Almost everywhere, a.e. Probabilists’ Term Probability space ( Ω , F , P ) Probability P : F → R ( σ -)field F on Ω Event E ∈ F Random variable X : Ω → R Expextation of f , E ( X ) X has finite p th moment Almost surely, a.s.
{ X > a } : = { w : X ( w ) > a } P ( X > a ) : = P ( { w : X ( w ) > a } )
\begin{align*}
\left\{ X \gt a \right\} &:= \left\{ w : X(w) \gt a \right\} \\
P\left( X \gt a \right) &:= P\left( \left\{ w : X(w) \gt a \right\} \right)
\end{align*}
{ X > a } P ( X > a ) := { w : X ( w ) > a } := P ( { w : X ( w ) > a } )
基礎定義 可測空間( Ω , F ) (\Omega, \mathcal{F}) ( Ω , F ) ( R , B R ) (\mathbb{R}, \mathcal{B}_{\mathbb{R}}) ( R , B R ) ( F , B R ) (\mathcal{F}, \mathcal{B}_{\mathbb{R}}) ( F , B R ) X : Ω → R X : \Omega \to \mathbb{R} X : Ω → R 確率変数 と言います。つまり、
X − 1 ( B ) ∈ F ∀ B ∈ B R .
X^{-1}(B) \in \mathcal{F}\qquad \forall B \in \mathcal{B}_{\mathbb{R}}.
X − 1 ( B ) ∈ F ∀ B ∈ B R .
( Ω , F ) (\Omega, \mathcal{F}) ( Ω , F ) 確率 (または確率測度)とは、P ( Ω ) = 1 P(\Omega) = 1 P ( Ω ) = 1 P : F → R P : \mathcal{F} \to \mathbb{R} P : F → R
X X X
期待値 : E ( X ) : = ∫ X d P \displaystyle E(X) := \int X dP E ( X ) := ∫ X d P 分散 : σ 2 ( X ) : = E [ ( X − E ( X ) ) 2 ] = E ( X 2 ) − E ( X ) 2 \sigma^{2}(X) := E\left[ (X - E(X))^{2} \right] = E(X^{2}) - E(X)^{2} σ 2 ( X ) := E [ ( X − E ( X ) ) 2 ] = E ( X 2 ) − E ( X ) 2 X X X (確率)分布 とは、次を満たすR \mathbb{R} R P X : B R → R P_{X} : \mathcal{B}_{\mathbb{R}} \to \mathbb{R} P X : B R → R P X ( B ) : = P ( X − 1 ( B ) ) .
P_{X}(B) := P(X^{-1}(B)).
P X ( B ) := P ( X − 1 ( B )) .
X X X 分布関数 F X F_{X} F X F X ( a ) : = P X ( ( − ∞ , a ] ) = P ( X ≤ a ) .
F_{X}(a) := P_{X}\left( (-\infty, a] \right) = P(X \le a).
F X ( a ) := P X ( ( − ∞ , a ] ) = P ( X ≤ a ) .
確率変数の数列{ X i } i = 1 n \left\{ X_{i} \right\}_{i=1}^{n} { X i } i = 1 n 確率ベクトル ( X 1 , … , X n ) (X_{1}, \dots, X_{n}) ( X 1 , … , X n ) ( X 1 , … , X n ) : Ω → R n
(X_{1}, \dots, X_{n}) : \Omega \to \mathbb{R}^{n}
( X 1 , … , X n ) : Ω → R n ( X 1 , … , X n ) ( x ) : = ( X 1 ( x ) , … , X n ( x ) ) .
(X_{1}, \dots, X_{n})(x) := (X_{1}(x), \dots, X_{n}(x)).
( X 1 , … , X n ) ( x ) := ( X 1 ( x ) , … , X n ( x )) .
Note: ( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) = X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n ) (X_{1}, \dots, X_{n})^{-1}(B_{1} \times \cdots \times B_{n})= X_{1}^{-1}(B_{1}) \cap \cdots \cap X_{n}^{-1}(B_{n}) ( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) = X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n )
n = 2 n=2 n = 2 ( X , Y ) : Ω → R 2 (X, Y) : \Omega \to \mathbb{R}^{2} ( X , Y ) : Ω → R 2 ( X , Y ) − 1 ( a , b ) = { x ∈ Ω : X ( x ) = a } ∩ { x ∈ Ω : Y ( x ) = b } .
(X, Y)^{-1} (a, b) = \left\{ x \in \Omega : X(x) = a \right\} \cap \left\{ x \in \Omega : Y(x) = b \right\}.
( X , Y ) − 1 ( a , b ) = { x ∈ Ω : X ( x ) = a } ∩ { x ∈ Ω : Y ( x ) = b } . B 1 B_{1} B 1 B 2 ∈ B R B_{2} \in \mathcal{B}_{\mathbb{R}} B 2 ∈ B R ( X , Y ) − 1 ( B 1 × B 2 ) = ( X , Y ) − 1 ( B 1 , B 2 ) = X − 1 ( B 1 ) ∩ Y − 1 ( B 2 ) .
(X, Y)^{-1}(B_{1} \times B_{2}) = (X, Y)^{-1}(B_{1}, B_{2}) = X^{-1}(B_{1}) \cap Y^{-1}(B_{2}).
( X , Y ) − 1 ( B 1 × B 2 ) = ( X , Y ) − 1 ( B 1 , B 2 ) = X − 1 ( B 1 ) ∩ Y − 1 ( B 2 ) . R n \mathbb{R}^{n} R n ( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) = ( X 1 , … , X n ) − 1 ( B 1 , … , B n ) = X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n ) .
\begin{equation}
\begin{aligned}
(X_{1}, \dots, X_{n})^{-1}(B_{1} \times \cdots \times B_{n})
&= (X_{1}, \dots, X_{n})^{-1}(B_{1}, \dots, B_{n}) \\
&= X_{1}^{-1}(B_{1}) \cap \cdots \cap X_{n}^{-1}(B_{n}).
\end{aligned}
\end{equation}
( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) = ( X 1 , … , X n ) − 1 ( B 1 , … , B n ) = X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n ) .
X 1 , … , X n X_{1}, \dots, X_{n} X 1 , … , X n 結合分布 とは確率ベクトル( X 1 , … , X n ) (X_{1}, \dots, X_{n}) ( X 1 , … , X n ) P ( X 1 , … , X n ) : B R n → R ,
P_{(X_{1}, \dots, X_{n})} : \mathcal{B}_{\mathbb{R}^{n}} \to \mathbb{R}, P ( X 1 , … , X n ) : B R n → R , P ( X 1 , … , X n ) ( B 1 × ⋯ × B n ) : = P ( ( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) ) .
P_{(X_{1}, \dots, X_{n})}(B_{1} \times \cdots \times B_{n}) := P\left( (X_{1}, \dots, X_{n})^{-1}(B_{1} \times \cdots \times B_{n}) \right).
P ( X 1 , … , X n ) ( B 1 × ⋯ × B n ) := P ( ( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) ) .
独立 P ( E ) > 0 P(E) \gt 0 P ( E ) > 0 E E E Ω \Omega Ω P E ( F ) = P ( E ∣ F ) : = P ( E ∩ F ) / P ( E )
P_{E}(F) = P(E|F) := P(E \cap F)/P(E)
P E ( F ) = P ( E ∣ F ) := P ( E ∩ F ) / P ( E ) E E E 条件付き確率 と言います。
もしP E ( F ) = P ( F ) P_{E}(F) = P(F) P E ( F ) = P ( F ) F F F E E E 独立 と言います：
F is independent of E ⟺ P ( E ∩ F ) = P ( E ) P ( F ) .
\text{F F F E E E F is independent of E ⟺ P ( E ∩ F ) = P ( E ) P ( F ) . Ω \Omega Ω { E j } \left\{ E_{j} \right\} { E j } P ( E 1 ∩ ⋯ ∩ E n ) = P ( E 1 ) P ( E 2 ) ⋯ P ( E n ) = ∏ i = 1 n P ( E j ) .
P(E_{1} \cap \cdots \cap E_{n}) = P(E_{1}) P(E_{2}) \cdots P(E_{n}) = \prod \limits_{i=1}^{n} P(E_{j}).
P ( E 1 ∩ ⋯ ∩ E n ) = P ( E 1 ) P ( E 2 ) ⋯ P ( E n ) = i = 1 ∏ n P ( E j ) .
Ω \Omega Ω { X j } \left\{ X_{j} \right\} { X j } 独立 ということは、全てのボレル集合B j ∈ B R B_{j} \in \mathcal{B}_{\mathbb{R}} B j ∈ B R { X j − 1 ( B j ) } \left\{ X_{j}^{-1}(B_{j}) \right\} { X j − 1 ( B j ) } P ( X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n ) ) = ∏ j = 1 n P ( X j − 1 ( B j ) ) .
P\left(X_{1}^{-1}(B_{1}) \cap \cdots \cap X_{n}^{-1}(B_{n})\right) = \prod \limits_{j=1}^{n} P(X_{j}^{-1}(B_{j})).
P ( X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n ) ) = j = 1 ∏ n P ( X j − 1 ( B j )) .
確率分布の定義と( 2 ) (2) ( 2 ) P ( X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n ) ) = P ( ( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) ) = P ( X 1 , … , X n ) ( B 1 × ⋯ × B n ) .
\begin{align*}
P\left(X_{1}^{-1}(B_{1}) \cap \cdots \cap X_{n}^{-1}(B_{n})\right)
&= P\left( (X_{1}, \dots, X_{n})^{-1}(B_{1} \times \cdots \times B_{n}) \right) \\
&= P_{(X_{1}, \dots, X_{n})} \left( B_{1} \times \cdots \times B_{n} \right).
\end{align*}
P ( X 1 − 1 ( B 1 ) ∩ ⋯ ∩ X n − 1 ( B n ) ) = P ( ( X 1 , … , X n ) − 1 ( B 1 × ⋯ × B n ) ) = P ( X 1 , … , X n ) ( B 1 × ⋯ × B n ) . ∏ j = 1 n P ( X j − 1 ( B j ) ) = ∏ j = 1 n P X j ( B j ) = ( ∏ j = 1 n P X j ) ( B 1 × ⋯ × B n ) .
\prod \limits_{j=1}^{n} P(X_{j}^{-1}(B_{j}))
= \prod \limits_{j=1}^{n} P_{X_{j}}(B_{j})
= \left( \prod \limits_{j=1}^{n} P_{X_{j}} \right) \left( B_{1} \times \cdots \times B_{n} \right).
j = 1 ∏ n P ( X j − 1 ( B j )) = j = 1 ∏ n P X j ( B j ) = ( j = 1 ∏ n P X j ) ( B 1 × ⋯ × B n ) . { X j } \left\{ X_{j} \right\} { X j } P ( X 1 , … , X n ) = ∏ j = 1 n P X j .
P_{(X_{1}, \dots, X_{n})} = \prod\limits_{j=1}^{n}P_{X_{j}}.
P ( X 1 , … , X n ) = j = 1 ∏ n P X j .
{ X j } \left\{ X_{j} \right\} { X j } { X j } \left\{ X_{j} \right\} { X j }
参考文献 Robert G. Bartle, The Elements of Integration and Lebesgue Measure (1995) Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications (1999) 