Questions tagged [monotone-class-theorem]

For questions related to the monotone class theorem.

Filter by
Sorted by
Tagged with
1
vote
1answer
44 views

Simple integrands dense in square integrable integrands

we defined $\Pi_2=\Pi_2(\{X_t\})$ as the class of predictable processes $\{H_t\}$ such that $||H||_x:=(E[\int_0^{\infty}H_s^2d[X]_s]^{\frac{1}{2}}<\infty$ and $\mathcal{M}^2$ as the set of all $L^...
0
votes
0answers
22 views

Apply monotone class theorem to define measures

I give you an example of what I mean in the title. I know the famous monotone class theorem, and I also know the Kolmogorov extension theorem. If I have a consistent family of finite-distributional ...
1
vote
0answers
24 views

monotone class theorem application, equality!

Ive been struggling trying to really learn what the monotone class theorem is about. I got the following problem: Let $(\Omega,F)$ a measurable space and $\mathbb{A}\subset F$ an algebra. Let $\mu$ ...
1
vote
0answers
38 views

$\mathbb{E}[f(X,Y)|Y]=\mathbb{E}[f(X,Y)]$ for all bounded measurable $f$ using the monotone class theorem

Let $X,Y$ be independent rv's. For $f:\mathbb{R}^2\to\mathbb{R}$ measurable we have $f_X:\mathbb{R}\to\mathbb{R}$ where $$f_X(y)= \left\{ \begin{array}{lr} \mathbb{E}[f(X,y)] & : \...
2
votes
2answers
21 views

A (sort of) converse to the monotone class theorem

In my notes we have the monotone class theorem which says that if a vector space of functions contains all the indicator functions of measurable sets and is closed under bounded monotone limits then ...
0
votes
1answer
72 views

Monotone Class Theorem for Bounded Nonnegative Functions

I'm doing an exercise regarding a version of monotone class theorems for functions. Let $\mathcal{C}$ be a collection of nonnegative bounded functions on $\Omega$. Then the following two statements ...
0
votes
0answers
7 views

Nonmonotone functions in compact convex space

I am asking for more explaining about : If we have a function $ F\mathrm{(}x\mathrm{)} $ which is not monotone in $ x $ And then we can say that the function : $ \overline{F}{\mathrm{(}}{x}{\mathrm{...
2
votes
0answers
74 views

Exercises on Dynkin Lemma and Monotone class Theorem

I am trying to understand and to apply two fundamental theorems in probability and measure theory: (a) Dynkin Lemma aka. $\pi/\lambda$ Theorem (b) Monotone class theorem W.r.t. (b) I had to solve ...
0
votes
1answer
631 views

Counterexample for Monotone convergence theorem [duplicate]

Let $f_n : [0,1] \rightarrow \mathbb{R}$ be a sequence of monotone decreasing measurable functions $f_n \geq f_{n+1}$ that converges pointwise to $f: [0,1] \rightarrow \mathbb{R}$. What would be the ...
2
votes
0answers
59 views

Related theorems of the monotone class theorem

The monotone class theorem in my measure theory text book is stated as follows. $\mathcal{C}$ is a family of sets. If $\mathcal{C}$ is an algebra, then $m(\mathcal{C}) = \sigma(\mathcal{C})$, ...
0
votes
0answers
113 views

Question about Monotone Class implies Sigma Field

I'm trying to solve this exercise from Resnick "A Probability Path" (1.41) and found the answer on the topic bellow, A field being a sigma field if and only if it's a monotone class Therefore, ...
4
votes
1answer
109 views

Monotone class theorem and measurability of random variables

Consider a family $f_i$, $i \in I$, of mappings of a set $\Omega$ into measurable spaces $(E_i,\mathcal{E}_i)$. We assume that for each $i \in I$ there is a subclass $\mathcal{N}_i$ of $\mathcal{E}_i$ ...
0
votes
1answer
144 views

Overkill? On the proof of Monotone Class Theorem in Probability Theory

I have studied the proof of Monotone Class Theorem from John B. Walsh's Knowing the Odds. I feel that a part of the proof is an overkill. I'm attaching the proof below. The author has defined $\...
3
votes
0answers
42 views

If an integrable function has a nonnegative integral over any set of a generating algebra, then it has a nonnegative integral over any measurable set

Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space $f\in\mathcal L^1(\mu)$ and $$\mathcal M:=\left\{A\in\mathcal A:\int_Af\:{\rm d}\mu\ge0\right\}$$ $\mathcal E\subseteq\mathcal M$ be ...
1
vote
1answer
67 views

Is the assumption of being an algebra crucial in the monotone class theorem?

I know the monotone class theorem as stated on Wikipedia: Let $G$ be an algebra of sets and define $\mathcal{M}(G)$ to be the smallest monotone class containing $G$. Then $\mathcal{M}(G)$ is ...
2
votes
1answer
41 views

Why the limit of a series of increasing/decreasing sets in a Monotonous class is usually described as their union or intersection?

When referring to the limit of a series of increasing/decreasing sets in Monotonous class, it is common to write this limit as the union/intersection of those sets. For example, when you read the ...
4
votes
0answers
209 views

Prove that $\mathcal{C}$ is a monotone class

Let $(\Omega_1,\mathcal{A}_1,\mu_1)$ and $(\Omega_2,\mathcal{A}_2,\mu_2)$ be two measure spaces, where $\mu_1$ and $\mu_2$ are $\sigma$-finite. Consider the product space $(\Omega=\Omega_1\times\...
3
votes
0answers
111 views

Monotone class definition

I've been presented with a definition of monotone classes that I was unfamiliar with : a subset of the power set containing the entire space, closed under countable increasing union and difference ...
0
votes
1answer
445 views

Monotonic property of solution of a linear (second-order) differential equation

Consider the following differential equation \begin{equation} y''+q(t) y=0 , \end{equation} where q(t) is a continuous function $\leq 0\;\; \forall t \in \mathbb{R}$. I'm trying to prove that each ...
5
votes
1answer
107 views

Kolmogoroff 0-1 does this proof work?

I have thought at this proof of the Kolmogorov 0-1 Law varying a little the sketch found in Probability essentials (Jean Jacod, Philip Protter). My questions are Is it a valid proof? Is it a bad ...
7
votes
1answer
1k views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
0
votes
1answer
42 views

Monotonicalli Sequence Theorem [closed]

Using the Monotonic Sequence Theorem prove that the following sequences are convergent. (𝑖) 𝑆𝑛=Σ 1/(𝑛+𝑘) (𝑖𝑖) 𝑆1=1, 𝑆𝑛+1=𝑆𝑛/√(𝑆𝑛2+1) , 𝑛∈ℤ+
2
votes
0answers
146 views

Extension of Cameron-Martin formula via monotone class theorem

my question revolves around the Cameron-Martin theorem: Let $(\mathcal{C}_{(0)}[0,1],\mathcal{B}(\mathcal{C}_{(0)}),\mu)$ be the Wiener space (i.e. continuous functions starting in $0$, equipped ...
1
vote
1answer
43 views

An inverse use of monotone class theorem

A family of sets $C$(or say a set of sets) is a monotone class if it is closed under countable monotone limit that is if $A_n ∈ C$, $n \geq 1$, $A_n \uparrow A$(or $A_n \downarrow A$) then $A∈C$ where ...
2
votes
1answer
65 views

Could monotone class theorem be applied to the proof of inequality (of probability)

I have 4 independent random variables: $X_1,X_2,Y_1,Y_2$. And I know for any measurable sets $S_1 \in \mathcal{F}_1, S_2 \in \mathcal{F}_2$, I have $$\Pr(X_1 \in S_1) \leq c\Pr(Y_1 \in S_1)$$ $$\Pr(...
8
votes
1answer
2k views

Monotone class theorem vs Dynkin $\pi-\lambda$ theorem

Monotone class theorem: Let $\mathcal C$ be a class of subset closed under finite intersections and containing $\Omega$ (that is, $\mathcal C$ is a $\pi$-system). Let $\mathcal B$ be the ...
2
votes
1answer
718 views

Smallest Monotone Class Subset of Smallest Sigma-Algebra?

$\newcommand{\A}{\mathcal{A}}\newcommand{\M}{\mathcal{M}}$ I'm having trouble understanding the first step of the Monotone Class Theorem, which every proof I find seems to claim is obvious. Here is ...
3
votes
0answers
74 views

can the emphasis on “smallest” in the monotone class theorem be ignored in applications?

The monotone class theorem states that for any algebra of sets $\cal A$ one can construct the smallest monotone class generated by this class ${\cal M}(\cal A)$. This smallest monotone class is also ...
5
votes
1answer
410 views

Monotone Class Theorem for Functions

Suppose $\mathcal F$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal F$ and $f + g$, $fg$, and $cf$ are in $\mathcal F$ whenever $f, g \in \mathcal F$ ...
3
votes
1answer
360 views

Proof using Monotone Class Theorem

As you know I have been grappling with this question since days ago, which I copy down here for convenience: Let $X$ be set of $\mathbb R$, and let $\mathcal B$ be its Borel $\sigma$-algebra, and ...
1
vote
1answer
65 views

Uniquness theorem from Monotone class theorem

I need to show using monotone class theorem (MCT) that; If $\mathcal{F}$ is a field, $P_1,P_2$ are two probability measures on $\sigma(\mathcal{F})$, then if $P_1=P_2$ on $\mathcal{F}$ then $P_1=P_2$ ...
1
vote
1answer
320 views

Monotone Class Theorem Application

I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} \mathbb{E}(h_{\...
0
votes
1answer
874 views

every $\sigma$ algebra is a monotone class

I couldn't understand the monotone class theorem because of this lemma: "Every $\sigma$ algebra is a monotone class." How i can prove it?
0
votes
1answer
88 views

monotone class theorem failure for a class of subsets that is not a field

Show the monotone class theorem fails if $F_{0}$ is not assumed to be a field. Monotone class theorem: Let $F_{0}$ be a field of subsets of $\Omega$, and $C$ a class of subsets of $\Omega$ that is ...
4
votes
1answer
3k views

monotone class theorem, proof

I am having difficulty with this proof: It is the three sentences I have colored that is very difficult. Could someone please explain why they are true? red line: I understand "Since A is an algebra"...
5
votes
1answer
188 views

Monotone class theorem for unbounded functions

So the statement for Monotone Class Theorem goes: Let $(\Omega,\mathcal{F})$ be a space with a $\sigma$-algebra $\mathcal{F}$ and with $\pi$-system $\mathcal{A}$ generating $\mathcal{F}$. If $ V $ ...
2
votes
0answers
299 views

Is there monotone class theorem used in one of these steps?

IN Rogers & Williams "Diffusions, Markov Process and Martingales" they introduce the resolvent as: $$R_\lambda f(x):=\int_{[0,\infty)}e^{-\lambda t}P_tf(x)dt=\int_ER_\lambda(x,dy)f(y)$$ where $...
3
votes
1answer
2k views

Proof of the Monotone Class Theorem

I am learning the Monotone Class Theorem from Jacod's Probability Essentials. I don't quite understand the idea of the proof in the book. I don't see the point in the proof at all. What's the use of ...
3
votes
1answer
933 views

A concrete example of the monotone class theorem

I have a lot of trouble in applying the functional monotone class theorem. Therefore I'm solving some exercises to get some experience. Maybe someone could help me with the following. Suppose I have ...
7
votes
1answer
905 views

Application of Monotone class Theorem in the proof of Kunita-Watanabe Inequality

The Kunita-Watanabe Inequality says: Let $X,Y$ be two continuous locale martingales and $H,G$ two product-measurable functions on $(0,\infty)\times \Omega$, then $$ \int_0^t|G_s||H_s|d|\langle X,...
13
votes
1answer
2k views

Monotone class theorem

I have some question about the Monotone Class Theorem and its application. First I state the Theorem: Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that $f_\...