All Questions
734
questions
0
votes
0
answers
28
views
Strict inequality of functions only allows to deduce a non-strict inequality of the expected value of said function
In a proof of Jensen's Inequality that I am reading, the following is used:
If for a real valued random variable $X$, we have $X(\omega)<\beta$, then $\mathbb{E}[X]\leq \beta$.
Why can we deduce ...
- 365
1
vote
0
answers
91
views
Differentiability of an integral depending on a parameter
Let $f$ be a function in $L^1 ([0,1])$, and for $y \in [0,1]$ consider $$F(y) = \int_{[0,1]} (1+ |f|)^y dx$$
Is $F$ differentiable in $(0,1)$? If it is, what is its derivative?
I know that that, given ...
- 11
0
votes
1
answer
46
views
If $A$ is Borel measurable, is $\int_0^21_{A}(x)\int_{x-4}^x\mathrm{d}u\mathrm{d}x=0$?
Consider the simple double integral $I(A)=\int_0^21_A(x)\int_{x-4}^x\mathrm{d}u\mathrm{d}x$ where $A$ is a set in the Borel $\sigma$-algebra over $\mathbb{R}$. I want to check the very simply question ...
- 362
0
votes
1
answer
40
views
Defining a measure on a finite dimensional Hilbert space
I am reading An Introduction to Infinite-Dimensional Analysis by Da Prato. Let $H$ be a $d$ dimensional Hilbert space where $d < \infty$ and $L^+(H)$ the set of symmetric, positive, and linear ...
- 6,277
4
votes
1
answer
544
views
Why are Lebesgue integrals defined as a supremum and not as a limit?
We may approximate a bounded nonnegative function $f$ by simple functions $\phi$. Then the standard way of defining the Lebesgue integral of $f$ is
$$ \int f d\mu := \sup \Bigg\{ \int \phi : \phi \...
- 6,277
1
vote
1
answer
61
views
$e^{-\lVert x \rVert ^2}$ is integrable over $\mathbb{R}^n$ and $\int_{\mathbb{R}^n} e^{-\lVert x \rVert ^2} = \pi^{n/2}$
$e^{-\lVert x \rVert ^2}$ is lebesgue integrable over $\mathbb{R}^n$ and $\int_{\mathbb{R}^n} e^{-\lVert x \rVert ^2} = \pi^{n/2}$. I'm a bit lost on this, I've tried to use a particular result about ...
- 800
0
votes
1
answer
55
views
"Leibniz's rule" for $t\in\mathbb{R}^n$
I am looking for a reference giving a measure-theoretic proof of a claim from the German Wikipedia. I have searched the references given on that site, as well as the English speaking Wikipedia and all ...
- 1
0
votes
0
answers
33
views
Use of dominated convergence theorem in Manski (1985)
I'm confused by the use of the dominated convergence theorem (DCT) in Lemma 5 of Manski (1985) (see below). Note that $b = (\tilde{b}_1, \dots, \tilde{b}_{K-1}, b_K).$
Specifically:
I presume the ...
- 147
0
votes
1
answer
26
views
Construction of a simple function by open sets
A simple function $\phi: \mathbb{R} \to \mathbb{R}$ can be written as:
$$\phi(x) = \sum_{k=1}^n a_k \chi_{E_k}(x)$$
where each $E_k$ is a measurable set and $\cup E_k = \mathbb{R}$ and $\chi_{E_k}$ is ...
- 751
0
votes
1
answer
104
views
Example of a function on $([0,1]; \sigma ([0;1]); \lambda )$ for which it has no meaning to write $ \int f d \lambda$
I am trying to understand Lebesgue integration and in order to understand well this concept I would like to have an example of a function $(0,1] $ ( or $ [0,1) , [0,1] , (0,1) $ ) for which it has no ...
- 719
0
votes
0
answers
62
views
Question concerning the correctness of this version of Fatou's Lemma
In lecture we learned about Fatou's Lemma stated as follows:
Let $(X, \mathcal{S}, \mu)$ be a measure space and $(f_k : X \to [0,\infty])$ measurable and $f: X \to [0, \infty] $ a function such that:
$...
- 615
0
votes
1
answer
156
views
Defining the expectation of a measurable function with respect to a (non-probability) measure
The typical definition of expectation requires a probability space and a random variable
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable ...
- 5,247
4
votes
1
answer
184
views
Confusion on "almost everywhere defined" function in $L^1$ space.
The following are excerpt from folland
This proposition shows that for the purposes of integration it makes no difference if we alter functions on null sets. Indeed, one can integrate functions $f$ ...
- 155
0
votes
1
answer
32
views
If a function $f : \Omega \to [0,\infty]$ is positive for uncountably many $x \in \Omega$, then counting measure integral on $\Omega$ is infinite
Let $(\Omega, 2^{\Omega}, \mu)$ be the measure space with $\mu$ the counting measure on a set $\Omega \neq \varnothing$ and $f : \Omega \to [0,\infty]$ a function. (All such functions are ...
- 1,445
0
votes
1
answer
37
views
A question on Monge formula- Optimal Transport
I have started reading optimal transport from the book "Optimal Transport
for Applied Mathematicians" and I have a question regarding change of variables in Monge's formulation.
We can ...
- 991