All Questions
Tagged with lattice-orders measure-theory
19
questions
3
votes
1
answer
46
views
Example of an infinite compact measurable space
Let $X$ be a nonempty set with a $\sigma$-algebra $\mathcal{A}$. The notion of $\sigma$-algebra strictly lies between Boolean algebras and complete Boolean algebras. Clearly, $\mathcal{A}$ is a ...
- 1,068
2
votes
0
answers
28
views
sigma-algebra vs sigma-frame
A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a family of subsets of $X$ such that:
$\phi\in \mathcal{A}$.
$\mathcal{A}$ is closed under countable unions.
$\mathcal{A}$ is closed under ...
- 1,068
9
votes
1
answer
508
views
Can the supremum of an uncountable family of measures be replaced by the supremum over a countable subfamily?
Consider a measurable space $(X,\mathcal{A})$. Let $\mathcal{M}$ denote the family of all countably additive measures $\mu\colon \mathcal{A}\to [0,+\infty]$. This family can be made into a partially ...
- 561
1
vote
1
answer
47
views
Proving a least upper property for $\sigma$-algebras on probability spaces
Let $(\Omega,\mathcal F,P)$ be probability space. Consider the partial order on $\mathcal F$ defined by $A\leq B \iff P[A\setminus B]=0$, i.e. almost sure inclusion. It has the following properties:
...
- 4,827
0
votes
0
answers
39
views
Lattice of self-adjoint bounded functionals on C*-algebra
When it comes to real-valued signed measures on a measurable space $(X, \mathcal{B})$, there’s a lattice structure given by
\begin{align*}
(\mu \lor \nu) (A) & = \sup \{ \mu (A \cap B) + \nu (A \...
- 8,769
2
votes
0
answers
74
views
Are these families the clopen sets of some class of topologies?
What would one call a family $\mathcal{F}$ of subsets $X$ such that:
$$(1):X\in\mathcal{F}$$
$$(2):A,B\in\mathcal{F}\implies A\setminus B\in\mathcal{F}$$
$$(3):P\subseteq\mathcal{F}\text{ is a ...
- 10.6k
1
vote
1
answer
68
views
Two different definitions of solid subspaces of a Riesz space
In D.H. Fremlin, Topological Riesz Spaces and Measure Theory, [14C], a subset $F$ of a Riesz space $E$ is defined as solid if
$x \in F$ whenever there is an $y \in F$ such that $|x| \le |y|$.
In G. ...
- 644
1
vote
1
answer
126
views
Prove that a solid linear subspace of a Riesz space is a Riesz subspace
In D.H. Fremlin, Topological Riesz Spaces and Measure Theory [14F] it is stated that
Let $E$ be a Riesz space. A Riesz subspace of $E$ is a linear subspace which is also a sublattice. A solid linear ...
- 644
2
votes
1
answer
212
views
Characterising $\sigma$-algebras as posets
A $\sigma$-algebra is defined as a set $X$ together with a subset $\Sigma$ of the power set $\mathcal{P}(X)$, such that
$X\in \Sigma$
$\Sigma$ is closed under complementation
$\Sigma$ is closed under ...
- 7,379
0
votes
0
answers
24
views
Lattice on $(M(X,\mathcal{A},\mathbb{R}),\leq)$
Show that $(M(X,\mathcal{A},\mathbb{R}),\leq)$ defines a lattice.
I know that $M(X,\mathcal{A},\mathbb{R})$ with the relation $\leq$ defines a partial order. Now, a lattice consists of a partially ...
- 700
2
votes
1
answer
188
views
Is the lattice of subspaces of a finite-dimensional scalar product space distributive?
A famous 1936 Birkhoff, Von Neumann paper claimed that the following lattice isn't distributive: the set $L(\mathscr{H})$ of closed linear subspaces of an Hilbert space $\mathscr{H}$, endowed with the ...
- 133
1
vote
1
answer
47
views
Sets equipped with sublattice of their power sets
Topology and measure theory are two examples of fields which include, in their objects of study, sets which are given structure by equipping them with a lattice of a certain type sitting in their ...
3
votes
1
answer
255
views
What is the Stone space of the free sigma-algebra on countably many generators
The Stone space of the free Boolean algebra on countably many generators is the Cantor space $2^\omega$. What is the Stone space of the free (Boolean) $\sigma$-algebra on countably many generators?
- 584
3
votes
0
answers
56
views
Counting the number of distinct weighted subsets
Suppose you have a finite set of items $X$. You want to assign a positive number $\mu(S)$ to each subset $S\subseteq X$ in such a way that you respect subset ordering:
$$S \subsetneq T \Rightarrow \mu(...
- 17.7k
1
vote
0
answers
63
views
Topological properties of the Join Operation of sigma algebras
Let $(\Omega,\mathcal{F},P)$ be a probability space. Is there a topology on the set of all sub sigma algebras of $\mathcal{F}$, denoted by $\mathcal{F}^*$, that makes the join operation continuous?
...
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