Tagged Questions
6
votes
2answers
109 views
It is possible to define our intuitive notion for probability in subsets of $[0,1]$
I've always heard and read the sentence:
If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$.
What is the meaning for that? Is this the "real" ...
17
votes
4answers
945 views
Tricks to remember Fatou's lemma
For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality
$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow ...
1
vote
1answer
118 views
What is a product $\sigma$-algebra?
My question is relatively simple: what is a product $\sigma$-algebra? And why they are important?
Can anyone suggest any links of intuitive (possibly with simple figures) explanations? Or, maybe ...
18
votes
1answer
424 views
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
10
votes
4answers
347 views
Why is Lebesgue integration better suited for convergence axioms?
I am trying to understand Lebesgue integration here. Here you basically make equal splits on the y-axis instead of the splits on the x-axis that Riemann does. I understand the proofs of the limit ...
6
votes
3answers
470 views
Significance of $\sigma$-finite measures
From Wikipedia:
The class of $\sigma$-finite measures has some very convenient properties;
$\sigma$-finiteness can be compared in this respect to separability of
topological spaces. Some ...
4
votes
1answer
251 views
What's the intuition behind and some illustrative applications of probability kernels?
Given measure spaces $(X, \mathcal{X})$ and $(Y, \mathcal{Y})$ we define measure kernel $\pi : \mathcal{X} \times Y \to [0,\infty]$ such that $\pi(\cdot|y)$ is a measure on $\mathcal{X}$ for every $y ...
28
votes
7answers
1k views
Why do we restrict the definition of Lebesgue Integrability?
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we ...
0
votes
0answers
142 views
Visualization of 2-dimensional function spaces
As a follow-up question to what is the norm measuring in function spaces
I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
0
votes
3answers
717 views
What is the norm measuring in function spaces
In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors.
The generalization to function spaces is quite a mental leap (at ...
