# Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

2,880
questions

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### Maximal element w.r.t. abolute continuity of measures

Suppose that $\mu$ is a $\sigma$-finite measure on $\mathcal{X}\equiv\bigotimes_{i=1}^n\mathcal{X}_i$. Let $\Pi$ denote the set of all $\sigma$-finite product measures on $\mathcal{X}$. Define
$$
\...

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0
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64
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### Some stability and estimate of the optimal transport map (Brenier map)

Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...

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### Conditional joint distribution decomposition

Let $I$ be a finite index set and $\mathcal X_i$ for $i\in I$ be a standard measurable space (or say a Polish space or $\mathbb R$). Suppose that the probability measure $P(X_I)$ on the product space $...

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43
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### Existence of derivative of distribution of exponential family?

Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is ...

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### An auxiliary problem while constructing the system of Jordan sets on a plane

Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...

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145
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### The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish

It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...

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223
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### Does a generic linear map admit a vector whose iterates span $V$?

We say a linear map $T$ on a finite dimensional vector space $V$ admits spanning vectors if there exists some vector $v \in V$ whose iterates $v, Tv, T^2 v, \dots$ under $T$ span $V$.
Question: ...

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1
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81
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### Lower bounds for truncated moments of Gaussian measures on Hilbert space

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...

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1
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84
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### Interchange the deterministic and stochastic integrals

We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \...

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### Existence of a measurable maximizer

Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...

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### Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?

Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...

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111
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### On a density property of signed singular measures

Suppose that $\mu$ is a signed finite Borel measure which is singular with respect to the Lebesgue measure in $[0,1]$. Is it true that there always exist a point $x\in [0,1]$ such that
\begin{equation*...

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1
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226
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### Is the Borel-Cantelli Lemma applicable here? [duplicate]

Consider $(X_{n})_{n\in\mathbb{N}}$ a sequence of random variables taking values in the set $\mathbb{Z}_{\geq 0}$ where $\mathbb{P}(X_{n} = i) > 0 $ for every $i\in\mathbb{Z}_{\geq0}$ which are ...

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### Worst of both worlds?

It's well known that $\mathsf{AC}$ implies the existence of non-measurable sets. And it's also true that, if all sets are measurable, then $|\mathbb{R}/\mathbb{Q}| > |\mathbb{R}|$. But is there a ...

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67
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### Same occupation measure $\Rightarrow$ same trajectory

Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
The occupation ...

18
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1
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342
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### Is defining measures as functionals ever insufficiently general in practice?

Crossposting from Math Stack Exchange, as it has yet to receive any answers there; the original question is here.
The way I learned measures was as set functions on a $\sigma$-algebra with certain ...

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1
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108
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### Some continuity issues of the optimal transport map (Brenier map)

Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...

1
vote

0
answers

77
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### Measurability of the union of cut loci along a curve

Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...

2
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0
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135
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### The Hausdorff measure of intersection of annulus and conformal curve

Recently I came across a problem in my research. Let $g:[0,1]\to\mathbb{C}$ be a restriction of a conformal map that is defined in a simply connected domain $\Omega\subseteq\mathbb{C}$ that include $[...

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1
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127
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### Strongly regular binary sequences

Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote the set of non-negative integers. If $n\in\mathbb{N}$ we let $[n] = \{0,\ldots,n-1\}$. For $A
\subseteq \mathbb{N}$ we let $$\mu^+(A) = \lim\sup_{n\to\infty}\...

2
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2
answers

293
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### Existence of the limit of periodic measures

Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...

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305
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### Disintegration measures and differential forms

Let $X$ and $Y$ be smooth oriented manifolds of dimension $m$ and $n$, and let $f\colon X\to Y$ a proper smooth map. There is a theorem called the "Disintegration Theorem" which says ...

2
votes

1
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82
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### Does left-invariance imply right-invariance for closed unimodular normal subgroups?

Suppose we have a locally compact group $G$ and a closed unimodular normal subgroup $N$.
Let $\mu$ be a Borel probability measure on $G$ which is left $N$-invariant.
Does it follow that $\mu$ is right ...

2
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2
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221
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### Preimage of null sets under a monotone increasing function

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...

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1
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94
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### Relation between the convergence of measures and the evolution of their supports

Let a sequence of Borel probability measures $\mu_n$ converges to $\mu$ in the weak*-topology. Are there any related results on the evolution of the support of $\mu_n$?
For example, in which case, the ...

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0
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63
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### Is a bounded measurable convex function above its interior lower semi-continuous convex envelope?

Let $E$ be a locally convex topological vector space, let $C$ be a convex set which matches the closure of its relative interior $\mathring C=\{ x\in C : \forall y\in C,\exists z\in C,~x\in\mathopen]y,...

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36
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### Weak convergence of Gibbs measures with converging energy functions

Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$
and let $Q$ be a fixed probability measure on $A$.
For each $\theta>0$, define the probability ...

1
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1
answer

92
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### Convergence of ODE with uniform $L^\infty \cap L^1$ bound on nonlinearity

Consider the IVP
$$
\left\{
\begin{aligned}
\frac{d}{dt} \Phi_n(t,x) &= f_n(\Phi_n(t,x)) && \forall t \in \mathbf{R}_+ \\
\Phi_n(0,x) &= x && \forall x \in \mathbf{R}
\end{...

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2
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242
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### An integral inequality?

Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$.
I want to show that if
$$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{$\star$}$$
then
$$ \int_{\...

1
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1
answer

111
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### Can functions with "big" discontinuities be in $H^1$?

How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...

11
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3
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### Every function on reals a sum of two surjective real functions?

From this question, and the answer thereof, we can see that every real valued function on reals is a sum of two injective functions. Is the same true if we replace injectivity by surjectivity.
For ...

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1
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122
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### Is a bounded convex function $g$ that is non-negative on this particular convex set also non-negative on the its closure?

Setup :
Let $S$ be the simplex on $\mathbb N$, i.e. the set of probability distribution on the natural numbers. Suppose we have $p\in S$ such that, for all $n\geq 1$, $p_n > 0$. For any $\emptyset\...

1
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1
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103
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### Properties of the relatively bounded probability distributions on the simplex over the natural numbers

Setup :
Let $S$ be the simplex over $\mathbb N$, i.e. the set of probability distribution over $\mathbb N$. Let $p\in S$ be such that $0 < p_n$ for all $n\in \mathbb N$. Let $H$ be the Hilbert ...

7
votes

2
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583
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### Countably representing all closed sets of positive measure

This may be a naive question, but I don't see an immediate argument.
Question: Does there exist a sequence $\{C_m\}_{m=1}^\infty$ of Borel subsets of $[0,1]$ with positive Lebesgue measure $|C_m|>0$...

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0
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93
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### If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions
Representation
Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$.
We call $\...

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0
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69
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### $f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...

3
votes

1
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137
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### Does $L^1$ boundedness and convergence in probability imply convergence in probability of the Cesaro sums?

Let $X_n$ be a sequence of random variables with uniformly bounded $L^1$ norm. Suppose $X_n$ converges in probability to $X \in L^1$.
Is it true that the Cesaro sums $Y_n := \frac{1}{n} \sum_{i = 1}^n ...

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0
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68
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### Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$

This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...

11
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0
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473
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### Are there 100 points that are part of every half-density part of the plane?

Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$?
I am deliberately being vague ...

2
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0
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136
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### $\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$

Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...

0
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86
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### Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?

(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...

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0
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56
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### What is a metric for weak convergence of finite measures on a non compact, complete and separable metric space?

Consider the set of finite positive measures on a complete, separable, but not compact, metric space $S$, endowed with the topology under which a sequence of finite positive measures $\{\mu_n\}$ ...

3
votes

1
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163
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### Continuity of conditional expectation

Let $X$ be a compact metric space, $\mu$ a Borel probability measure on $X$ and $f: X \to \mathbb{R}$ a continuous function. Consider an increasing sequence of $\sigma$-algebras $A_n$ so that for all $...

1
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1
answer

110
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### In the limit, do the intersection points of a string figure define a probability measure on the unit disk?

Let D = {z ∈ ℂ | |z| ≤ 1} denote the closed unit disk in the complex plane.
For any integer n ≥ 1 define the nth string figure S(n) ⊂ D as the union of all n(n+1)/2 line segments that connect two ...

0
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0
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36
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### Measurability of the weak completion of an orthogonal representation

Let $G$ be a locally compact group and let $\pi$ be a strongly continuous orthogonal representation of $G$ in a real Hilbert space $H$. Denote by $E$ the real Hausdorff locally convex space obtained ...

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votes

1
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223
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### Book Reccomendation to learn measure theory? [closed]

I would like to know if there are any 'accessible' books to understand measure theory. I came across videos of Jay Cummings on YouTube channel 'The Bright Side of Mathematics' (if anyone knows that) ...

0
votes

1
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70
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### Existence and uniqueness of a posterior distribution

I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...

3
votes

0
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173
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### Reverse-mathematical strength of Banach-Tarski

What is the reverse mathematical strength of the Banach-Tarski paradox?
The usual proof of Banach-Tarski should carry out in $\mathrm{ZF}+\mathrm{AC}_\kappa$, where $\kappa$ is the supremum of the ...

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votes

1
answer

75
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### Pointwise limit of a "net" of measurable functions is measurable? [closed]

Let $(X, \mathcal{A},\mu)$ be a finite measure space with the $\sigma$-algebra $\mathcal{A}$ and the measure $\mu$.
Let $B$ be a separable Banach space. Then, it is well-known from a theorem by Pettis ...

1
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0
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77
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### Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...