Questions tagged [monotone-class-theorem]

For questions related to the monotone class theorem.

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Two equivalent versions of monotone class theorem

Let $\Omega$ be a set. Let $a(C), \lambda(C), m(C)$ be the smallest algebra, the smallest $\lambda$-system, and the smallest monotone class that contain $C$ respectively. Then we have two versions of ...
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Dynkin's $\pi-\lambda$ theorem is equivalent to monotone class theorem

Let $\Omega$ be a set. Let $a(C), \lambda(C), m(C)$ be the smallest algebra, the smallest $\lambda$-system, and the smallest monotone class that contain $C$ respectively. Dynkin's $\pi-\lambda$ ...
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The collection $\lambda(C)$ is a monotone class. If $C$ a $\pi$-system, then $a(C) \subset \lambda (C)$

Let $\Omega$ be a set. Let $a(C), \lambda(C), m(C)$ be the smallest algebra, the smallest $\lambda$-system, and the smallest monotone class that contain $C$ respectively. In proving that the monotone ...
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Let $G$ be an algebra. Then the smallest monotone class containing $G$ coincides with the smallest $\lambda$-system containing $G$

We fix $\Omega$ and let $G$ be an algebra on $\Omega$. I'm trying to prove below result. Could you please have a check on my attempt? Theorem: Let $M(G)$ be the smallest monotone class containing $G$ ...
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$(X,\Sigma,\mu)$ be a probability space, $A\subset\Sigma$ be a field s.t. $\Sigma=\sigma(A)$.Prove inf$\{\mu(E\Delta F):F\in A\}=0\forall E\in\Sigma$

$(X,\Sigma,\mu)$ be a probability space, $\mathcal{A}\subset\Sigma$ be a field such that $\Sigma=\sigma(A)$. Prove that, for all $E\in\Sigma$ $$\text{inf }\{\mu(E\Delta F):\ F\in \mathcal{A}\}=0$$ I ...
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Using the monotone class argument to prove that the set of transport plans is closed in the weak topology

i'm having some problem understanding the first answer given to the following question: Proof that the set of transference plans is closed in the weak topology. In particular i can't prove the second ...
3 votes
2 answers
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Find a family $\mathscr{F}$ such that $\mathscr{F}\subset \mathcal{M}\subset{\Phi_\mathscr{F}}$ with proper subsets.

Find a family $\varnothing\neq\mathscr{F}\subset \mathcal{P}(X)$ where $\Phi_\mathscr{F}$ is the $\sigma-$algebra generated by $\mathscr{F}$, $\mathcal{M}_\mathscr{F}$ the monotone class that contains ...
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monotone class theorem when $\Omega\not\in\mathcal A$, but $\Omega=\bigcup_{n\in\mathbb N}Ξ©_n$ for some $(Ξ©_n)_{n\in\mathbb N}\subseteq\mathcal A$

Remember the following monotone class theorem: Let $\Omega$ be a set; $E$ be a normed $\mathbb R$-vector space; $\mathcal A\subseteq2^\Omega$ be a $\pi$-system with $\Omega\in\mathcal A$; $\mathcal H\...
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Sigma algebra generated by bounded continuous functions on $\mathbb{R}$

Let $C_b(\mathbb{R})$ be the set of real-valued bounded continuous functions on $\mathbb{R}$. The $\sigma-$algebra generated by $C_b(\mathbb{R})$ is the smallest $\sigma-$algebra, say $\sigma(C_b(\...
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Absolute continuity of a measure in the intervals implies absolute continuity of borel measures?

Suppose I have two positive Borel measures on $[0,\infty]$, say $\mu$ and $\nu$ and I know that $\nu([0,x]) = 0$ implies $\mu([0,x]) = 0$ for any $x$ and that $\nu((a,b]) = 0$ implies $\mu((a,b]) = 0$ ...
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Procedure to show martingale property

When one's trying to prove the martingale property (assuming it's adapted and integrable), why does it suffice for a stochastic process $(X_t)_{t\geq0}$ satisfy $$ \mathbb{E}[(X_t-X_s)\prod_{i=1}^n ...
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Using the monotone class argument to prove martingale property

I'm having trouble with the following statement: For the stochastic process $(M_t)_{t\in[0,T]}$ to be a martingale it suffices to show that \begin{equation}\label{green} \mathbb{E}[(M_t-M_s)\prod_{i=1}...
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Extending measure on Field to $\sigma$-Field

Let $\mu_1$ and $\mu_2$ be two finite measures defined on $\sigma(\mathcal{F})$ such that, $\forall A \in \mathcal{F}$, $\mu_1(A)=\mu_2(A)$. Show that they must agree on $\sigma(\mathcal{F})$. I ...
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Compute inf and sup of a set A

Compute the infimum and supremum of $A=\{f(x)=\frac{2x+1}{x+2}: x>-2\}$. I try to do these following passages: since $f$ is derivable I compute $f'(x)=\frac{3}{x+2}>0$ and so from Monotone ...
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No-monotone sequence

A no-monotone sequence can have limit? Can I consider as no-monotone sequence $a_n=\frac{(-1)^n}{n}$?
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Show that ${ \mu(A \times B)=\mu_1(A) \cdot \mu_2(B) }$

I need help with a task. Let ${ X_1 }$${ X_2 }$ sets and ${\mathscr{A_i} \subset \mathcal{P}(X_i)}$ two algebras on ${X_i}$ for i=1,2. Let also ${ \mu_i: \mathscr{A_i} \to [0, \infty ] }$ be 2 ...
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Monotone convergence theorem for a decreasing recursive sequence from $n>n_0$

Let a recursive sequence: $$a(0)=a\in\mathbb{R};\\ a(n+1)=f(a(n));\\$$ if the sequence is decreasing but not for all $n\in\mathbb{N}$, e.g $a(n+1)\leq a(n) \,\, \forall \, n\geq 2$, it holds the ...
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Confused with the notation used in proof of monotone class theorem and the logic behind the proof

We were introduced to the monotone class theorem which stated that (in my own understanding) the minimal sigma field generated by the field $\mathcal{C}$ (denoted $\sigma[\mathcal{C}]$) equals to the ...
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How to prove that $\mathbb{E}Y1_A = \mathbb{E}X1_A, \forall A \in G \iff \mathbb{E}YZ = \mathbb{E}XZ, \forall Z \in L^\infty(G)$

I am trying to prove this statement using the monotone class theorem for functions : $\mathbb{E}Y1_A = \mathbb{E}X1_A, \forall A \in G \iff \mathbb{E}YZ = \mathbb{E}XZ, \forall Z \in L^\infty(G)$ I ...
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Monotone class theorem for semi algebras

Let $\Omega$ be a non empty set, and $\mathcal{A}$ be an algebra of sets of $\Omega$. Then, the Monotone Class Theorem asserts that the generated $\sigma$-algebra $\mathcal{F}(\mathcal{A})$ coincides ...
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Direct proof that integral of a function does not depend on the $\sigma$-algebra used to define it?

If $\mathcal{G}\subset\mathcal{F}$ are two $\sigma$ algebras on a set $X$, $\mu$ is a nonnegative measure on $(X,\mathcal{F})$ and $f:X\to[0,+\infty]$ is $\mathcal{G}$-measurable, then there are two ...
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Question about simple detail in the proof of the Monotone Class Theorem

My question relates to this previous thread: monotone class theorem, proof This thread provides a proof for the Monotone Class Theorem, using the following: Here $\mathcal{A}$ is an algebra and $\...
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How to prove that this set is a monotone class?

Monotone Class Definition: We say that $\mathcal{G}$ is a monotone class if whenever $\{A_{k}\}$ is an increasing and $\{B_{k}\}$ is a decreasing sequence in $\mathcal{G},$ then $\cup A_{k}$ and $\...
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A question about an example of a monotone class.

I am asking about this question: Checking an example of a monotone class. 1-Why in the example of the OP $(-\infty, a) \cup [a, \infty)$ not equal to $\mathbb{R}$? 2-How can I prove rigorously ...
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Checking an example of a monotone class.

Give an example of a monotone class $\mathcal{G}$ on $\mathbb{R}$ that satisfies: (a) \mathbb{R} belongs to $\mathcal{G},$ and (b)if $A \in \mathcal{G} $ then its complement $A^c$ is in $\mathcal{G},$...
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Independence from finite independence by Monotone Class Lemma

From the appendix of J.-F. Le Gall: Brownian Motion, Martingales, and Stochastic Calculus Here are a few consequences of the monotone class lemma that are used above. Let $(X_i : i \in I)$ be an ...
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Determine if the following sequence is monotonic

Consider the recursion $$ X_1 = 10 \quad \&\quad X_{n+1} = \sqrt {3+2X_n}$$ Is it monotonic or not? problem What I have done so far. $X_1 = 10$ $X_2 = \sqrt {23}$ $X_3 = \sqrt {2 \sqrt{23} +...
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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^...
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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 $\mu$ ...
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$\mathbb{E}[f(X,Y)|Y]=\mathbb{E}[f(X,Y)]$ 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)] & : \...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 $m(\mathcal{C}) = \sigma(\mathcal{C})$, ...
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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, ...
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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 $\mathcal{E}_i$ ...
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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 $\...
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3 votes
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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 ...
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1 answer
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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 ...
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2 votes
1 answer
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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 ...
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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\...
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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 ...
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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
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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 ...
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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 $\...
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Monotonicalli Sequence Theorem [closed]

Using the Monotonic Sequence Theorem prove that the following sequences are convergent. (𝑖) 𝑆𝑛=Ξ£ 1/(𝑛+π‘˜) (𝑖𝑖) 𝑆1=1, 𝑆𝑛+1=𝑆𝑛/√(𝑆𝑛2+1) , π‘›βˆˆβ„€+
2 votes
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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 ...
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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 ...