Advanced Probability Theory

Proposition:

Kolmogorov ineq. : Let be an i.i.d r.v. seq., , then , .
Supremum inequality, is a martingale

Theorem.: Let be a iid r.v. and , then converges a.e. .

Proof: W. L. O. G. , suppose . Let . converges a.s. as ,

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Definition: A measurable space is called separable, if every atom of it is singleton. Two measurable spaces are called isomorphic if there exists a bijection that is measurable both sides (which is called measurable a isomorphism). /%E5%88%9D%E7%AD%89%E6%8F%8F%E8%BF%B0%E9%9B%86%E5%90%88%E8%AE%BA/%E6%A0%87%E5%87%86Borel%E7%A9%BA%E9%97%B4)

Lemma: Let be a divisible and separable measure space, then is isomorphic to ’s some measurable subspace. Precisely, suppose is the algebra generating on . Let , then is a isomorphism from to . Here, .

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Monotone class theorem.’s:

Definition: Let be a algebra on . We call a measurable space and element of measurable set. If there is a countable sub family of s.t. , we call algebra is divisible. If is divisible, we call is a divisible measurable space.

Theorem1.2.3: Let be a family of sets.
(1) )
(2) .

Theorem.: If is divisible, then field which contains at most countable elements, and .

Proof: Let , . Then is a semi-field. Thus is a field. And .

Definition: Let is a measurable space, for any , let

$$\mathcal F_\omega = \{B \in \mathcal F \vert \omega \in B\}, A(\omega) = \bigcap_{B \in \mathcal F_\omega} B.$$

is call a atom containing .

Claim:

1. Let , then either or .
2. Let be the algebra generating . Then , let $\mathcal C_\omega = \{B \in \mathcal C \vert \omega \in B\}, then $$A(\omega) = \bigcap_{B \in \mathcal C_\omega} B.$$

Proof of 2:
For any given , let . Then . And is a monotone class. Thus , i.e. for any given , and . Hence . Hence .

Theorem: Let be a measurable space, is a measurable function.

1. There exists a sequence of simple function , s.t. , , and .
2. If is nonnegative, then there exists a monotonely increasing sequence of nonnegative simple measurable functions s.t. .

Proof: Writing as , it’s easy to see that 1. follows from 2.. , let

$$f_n = \sum_{k = 0}^{n2^n - 1} \frac{k}{2^n} \chi_{\{\frac{k}{2^n} \leq f < \frac{k + 1}{2 ^n}\}} + n \chi_{\{f \geq n\}}.$$

Then is a sequence of nonnegative simple measurable functions s.t. .

Theorem: Let be a measurable space, is a algebra generating . Let be a family of measurable function on . If:

1. ;
2. or ;
3. ,
   Then containing all -measurable function on .

Proof:
Let , then by 3. we know that , and by 2. we know that is a monotone class. Thus by monotone class thm. .

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Theorem: Let be a -system on , be a linear space consisting of some real-valued function on . If:

1. ;
2. , and is measurable(correspondingly, real-valued or bounded) ;
3. .
   Then containing all -measurable (correspondingly, real-valued or bounded) funcions on .

Proof: Let , then it’s obvious that is a -system and . Then by monotone class thm. . Suppose is -measurable real-valued function. Let

$$g_n = \sum_{k = 0}^{n2^n - 1} \frac{k}{2^n} \chi_{\{\frac{k}{2^n} \leq f^+ < \frac{k + 1}{2^n}\}} + n \chi_{\{f^+ \geq n\}},$$

Then , , then by 2., . By similar argument, . So . Q.E.D.

Theorem: Let be a family of real-valued function.

• If a null set . s.t. , , then we call that is almostly converges to . Denote it as a.e. or .
• If > , in , s.t. converges to on uniformly. Then we call converges to almost uniformly, denoted as .
• If > , . Then we call converges to in probability, denoted as .