# Questions tagged [monotone-class-theorem]

For questions related to the monotone class theorem.

67 questions
Filter by
Sorted by
Tagged with
29 views

### 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 ...
• 11.5k
1 vote
17 views

### 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 $\pi-\lambda$ ...
• 11.5k
18 views

### 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 ...
• 11.5k
23 views

### 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 $G$ ...
• 11.5k
1 vote
28 views

### $(X,\Sigma,\mu)$ 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 ...
• 515
11 views

### 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 ...
39 views

### 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 ...
• 2,101
1 vote
37 views

• 375
32 views

### 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 $\mu((a,b]) = 0$ ...
106 views

• 1,238
143 views

1 vote
71 views

• 1,461
133 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 ...
• 1,199
830 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 ...
137 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 ...
• 1,036