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Results tagged with measure-theory
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user 362671
A field of mathematics concerning measures, which are functions mapping sets to real numbers that generalize the idea of area and volume. Measure theory underlies modern treatments of probability. Questions about measure theory with a weak connection to probability or statistics may be more suitable for math.stackexchange.com than this site.
3
votes
Accepted
Kolmogorov axioms consequences
the prof mentioned that there does not exist a probability measure P(A) defined for all subsets of [0,1] which would satisfy all the 3 Kolmogorov axioms
At the very outset, please note whuber's comm …
- 10.3k
2
votes
Is $Y=Y(\omega) = \inf_{0 \leq t \leq 1}X_t(\omega) = 1_A(\omega)$ not measurable if $A \not...
Observation $1$. Let $(X, \mathfrak A) $ be a measurable space and let $A\subset X. $ Then $\mathbf 1_A$ is $\mathfrak A$-measurable if and only if $A\in \mathfrak A. $
The proof is straightforward ap …
- 10.3k
0
votes
Support of a continuous distribution
The most general definition that I am aware of takes support of any Borel measure $\mu$ on a topological space $X$ to be the the smallest closed set $A$ such that
$\mu(X\setminus A) = 0$ (cf. $\rm [I] …
- 10.3k
1
vote
Accepted
Inverting a characteristic function if the integral of the modulus of the cf is infinity
It doesn't make sense to use $f(x)=\frac{1}{2\pi}\int \exp(-\mathrm itx) \varphi_X(t)~\mathrm dt$ when $\int|\varphi_X(t) |~\mathrm dt=\infty, $ for the former is true if $\int|\varphi_X(t) |~\mathrm …
- 10.3k
4
votes
Finite additive probability defined on a "finite-additive" field
I think what you seek is premeasure. More formally, if $\mathcal S$ is any collection of subsets of $X$ then $\mu:\mathcal S\to [0, \infty]$ is a premeasure if it is finitely additive and countably mo …
- 10.3k
3
votes
Accepted
Confirmation of MGF of Shifted Exponential Distribution
What is the effect of change of origin and scale on moment generating function? Specifically if $X\mapsto \frac{X-a}{h}=: U, $ then what is $M_U(t) $ and is it related to $M_X(t) $ in any way?
The obs …
- 10.3k
5
votes
If $F_X, F_Y$ agree for all $x \in \mathbb{R}$, Do their distributions $\mu_X, \mu_Y$ agree ...
Observation $1.$ Let $\mathbf P_1, ~\mathbf P_2$ be two probability measures on $(\Omega, \mathcal F). $ Let $\mathcal P$ be a $\pi$-system such that $$\mathbf P_1(A) =\mathbf P_2(A), ~~~\forall A\in …
- 10.3k
3
votes
Pushforward measure for Radon Nikodym equation
This is a basic application of integration of image measure. Formally, this can be stated in the following way:
Result: Let $(\Omega_1,\mathscr F_1,\mathbb P) $ be a probability space. Let $\mu\ll \ma …
- 10.3k
8
votes
Accepted
Marginalisation with respect to arbitrary distribution
This is a classical set-up as can be formally stated as the Product Measure Theorem: if $(\Omega_1, \mathscr F_1) $ and $(\Omega_2, \mathscr F_2) $ are measurable spaces and $\mu_1$ is a $\sigma$–fini …
- 10.3k
3
votes
Accepted
"Almost surely" used in an expectation
In a measure space $(\Omega, \boldsymbol{\mathfrak{A}}, \mu) ,$ let $f$ be a nonnegative extended real-valued $\boldsymbol{\mathfrak{A}}$-measurable function on a domain set $D\in\boldsymbol{\mathfrak …
- 10.3k
7
votes
Two Different Proofs of Continuous Mapping Theorem
Zhanxiong has already elaborated on what Durrett is up to and what the Wikipedia article missed.
However let me emphasize the fact that the application of (Baby) Skorohod Theorem rather is ingenious a …
- 10.3k
1
vote
Density from characteristic function: Durrett example 3.3.8 and 3.3.9
For $\mathrm{Tri}(a) $ distribution (support on $[-a, a]$), the characteristic function is $$\varphi_{\mathrm{Tri}(a)}(t) =2\left[\frac{1-\cos(at)}{a^2t^2}\right], $$ the density being
$$f(x;a)= \frac …
- 10.3k
6
votes
References to learn « Real and Functional Analysis for Statisticians »
To study books like that of Shao's, Keener's or some other books of similar breed, you need to have a robust concept of not only real analysis but measure theory, general topology and functional analy …
- 10.3k
3
votes
Expectation conditional on a sigma algebra, what expectation does it refer to?
Apart from the extensive discussion in Zhanxiong's post, one can rely on this general, simple and yet useful result (which has been applied in the said answer) formally stated below:
Result $1.$ Let $ …
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