5
votes
1answer
236 views

Minkowski Content

Could someone provide some intuition behind the $n$-dimensional Minkowski Contentthe $n$-dimensional upper Minkowski Content of $\mathcal{A}$ as $$\mathfrak{M}^{*m} (\mathcal{A}) : = \lim_{\epsilon ...
0
votes
2answers
87 views

Lebesgue Measure Definition

Given a subset $A \subset \mathbb{R}$ with the length of an open interval $\mu_L(I_k) = b_k -a_k : I \doteq [a_k,b_k]$ The lebesgue measure is defined as $$ \lambda^{\ast} (A) \doteq \inf \Big\{ ...
4
votes
0answers
165 views

Intuition behind the definition of a measurable set

This week I saw the definition of a measurable set for an outer measure. Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + ...
6
votes
1answer
295 views

Motivation behind introduction of measure theory

Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a ...
11
votes
3answers
382 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" ...
26
votes
4answers
4k 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 ...
2
votes
1answer
401 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 ...
19
votes
1answer
588 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 ...
12
votes
4answers
518 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 ...
7
votes
3answers
894 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
296 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 ...
30
votes
7answers
2k 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
221 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
1k 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 ...