All Questions
187
questions
0
votes
0
answers
32
views
Is there a simple proof that the Laplace transform of a signed measure $\mu$ on $\mathbb R$ uniquely determines $\mu$? [duplicate]
A Laplace transform of a signed measure $\mu$ on $(\mathbb R, \mathcal B(\mathbb R))$ is defined by
$$ f_\mu(s) = \int_{\mathbb R} e^{-\lambda t} d \mu(t), \qquad \forall s \in \mathbb R. $$
I know ...
0
votes
1
answer
59
views
Estimating integrals and measures over Hilbert space using finite dimensional projections
Let $H$ be a separable Hilbert space with orthonormal basis $\{e_k\}_{k \in \mathbb{N}}$. Let $P_n$ be the projection onto the $n$-dimensional subspace of $H$:
$$P_n x = \sum_{i=1}^n \langle x, e_i\...
- 6,277
1
vote
0
answers
38
views
Layer cake representation and function with compact support
My question is related to the posts here and here, but my setup is slightly different.
For a real-valued random variable $X$, and a function $\varphi: \mathbb{R} \to \mathbb{R}$ that has support in ...
- 361
0
votes
1
answer
34
views
Tailsum Formula and Indicator Functions
In my probability theory class we proved that $$\mathbb{E}[x]=\int_0^\infty \mathbb{P}(X>t) dt,$$ where $X\geq0$ is a non-negative random variable and $\mathbb{E}[X]:= \int_\Omega X(\omega) d\...
- 325
1
vote
0
answers
39
views
Is every probability mass function $f_X$ of a random variable $X$ the Radon-Nikodym derivative of $X_*P$ with respect to the counting measure?
Let $X$ be a discrete random variable (meaning $\text{im}(X)$ is countable) from a probability triple $(A,\mathcal{A},P)$ to a measurable space $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the ...
- 5,166
0
votes
0
answers
11
views
Integral of a function with respect to a measure on a mixed space $[\![ N ]\!]\times \mathsf{X}$
Task
$\newcommand{\intbrackets}[1]{[\![ #1 ]\!]}$
$\newcommand{\Pcal}{\mathcal{P}}$
I would like to better understand integration of a function $f$ with respect to a probability measure $\mu$ on a ...
- 5,247
1
vote
2
answers
92
views
How to generalise inner product to measures without densities
Let $(E, \mathcal{E}, \lambda)$ be a metric finite measure space, and let $\mu, \nu$ be finite measures with densities $f,g$ with respect to $\lambda$.
Then, I am interested in considering the ...
- 2,466
0
votes
1
answer
52
views
Example of sequence of integrable maps such that integral smaller than $1/n$ but doesn't converge to zero almost everywhere
I am looking for a measure space $(\Omega,\mathcal{M},\mu)$ and a sequence of integrable functions $(f_n)$ in $\mathcal{L}^1(\Omega,\mathcal{M},\mu)$ with the property that
$\int_\Omega |f_n|d\mu \leq ...
- 415
5
votes
1
answer
279
views
First fundamental theorem of Calculus continuity not necessary?
I know if $f:[a,b] \to \mathbb{R}$ is continuous, then $g(x) = \int _{a}^{x}f(t) \, dt$ is differentiable on $[a,b]$. Furthermore, $g'(x) = f(x)$.
This is known as the first fundamental theorem of ...
- 1,706
1
vote
2
answers
32
views
How to calculate probability of outcome of nonhomogenous Poisson process?
Suppose I have a nonhomogenous Poisson process with a known rate function $r(t)$ over a time window $[0,T]$. Now suppose I use this process to generate events and perfectly measure the arrival time of ...
- 23
3
votes
2
answers
129
views
Does the collection of bounded continuous functions characterize probability law?
Let $X,Y :(\Omega,\mathcal{A},\mathbb{P}) \to \mathbb{R}$ be two r.v.'s defined on a probability space $\Omega$, and $C_b(\mathbb{R})$ the collection of all real bounded continuous function. I'm ...
- 782
2
votes
1
answer
84
views
Does this condition about two measures $p$ and $q$ imply existence of $p$-integrable function that is not $q$-integrable?
Context :
I am trying to characterize some properties of barycenters of measures on a probability space, the following question arises. More precisely I want to restrict the support of a measure on ...
- 6,076
1
vote
0
answers
87
views
Iterative Integration over indicator function of two variables
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X, Y$ random variables, mapping from said space to $(\mathcal{X}, \mathcal{C})$ and $(\mathcal{Y}, \mathcal{D})$, respectively.
The ...
- 365
0
votes
1
answer
35
views
Using Scheffé's lemma to show convergence of CDFs
Copying from this question, let $(S, \Sigma, \mu)$ be a measure space. (Part of) Scheffé's lemma states:
Suppose $\{f_n\}_{n \in \mathbb{N}}, f \in \mathscr{L}^1 (S, \Sigma, \mu)$ and $\lim_{n \to \...
- 367
1
vote
0
answers
31
views
Find $f_n,h_n$ such that $\lim_{n\to\infty}\int_{[0,1]\times\Omega\times[0,1]}(h_n(r,y)f_n(x)-X(r,y,x))^2d(\lambda\otimes P\otimes\lambda)(r,y,x)=0$
We consider a probability space $(\Omega,\mathcal{F},P),$ and for $(r,\omega,x) \in [0,1] \times \Omega \times [0,1],$ a bounded $\mathcal{B}([0,1]) \otimes \mathcal{F} \otimes \mathcal{B}([0,1])$-...
- 616