1
vote
0answers
13 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
3
votes
0answers
61 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
1
vote
0answers
53 views

Different definitions of uniform integrability

In different books and resources, I saw different definitions of uniformly integrable. For example, in some books the definition is like: Definition 1: $\{f_k\}\in L^1(E)$ is uniformly integrable ...
0
votes
1answer
13 views

Equivalence between submodular function definitions.

I am trying to show that the definitions, given by wikipedia, of a submodular set function are equivalent. See section definition of: http://en.wikipedia.org/wiki/Submodular_set_function. Mainly I ...
2
votes
0answers
43 views

Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
0
votes
1answer
24 views

Question about measure on set that is not in $\sigma$-algebra

I think I have problem with badly written book, or I just can't understand statement. Let $(X,\mathcal{A},\mu)$ be any measure space and $\mu^*$ be outer and $\mu_*$ inner measure inner measure ...
2
votes
0answers
57 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
1
vote
1answer
130 views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
0
votes
1answer
70 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
0
votes
0answers
64 views

Maximum measurable collection and Caratheodory extension

Remember Caratheodory extension theorem: we get an outer measure by extending a premeasure over an algebra. We also have Caratheodory theorem, which give us a measure by restricting that outer measure ...
1
vote
1answer
87 views

What sets are Lebesgue measurable?

I cannot detect the fallacy in the set of the following statements in my inconsistent notes: A sigma algebra is a set of the sets in the generating set closed under the set operations countable ...
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\{ ...
2
votes
1answer
63 views

Differing definitions for 'Algebra of subsets'

For a collection, $A$ of subsets of a set $X$ to be an algebra of subsets it must satisfy the following properties: $A$ is non-empty If $E \in A \implies E^c \in A$ If $E, F \in A \implies E \cup F ...
1
vote
3answers
110 views

Why is $\infty-\infty$ undefined in measure theory?

Some additions to the title: I stumbled over this problem going through my measure theory lecture notes; the author explicitly mentions that he leaves $\infty-\infty$ undefined. I would like to know ...
1
vote
2answers
157 views

Definition of Lebesgue-Stieltjes measure on $\mathbb R$

Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure $$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$ ...
11
votes
3answers
375 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" ...
0
votes
1answer
57 views

Difference in reference of L space in Fubini Tonelli

I think my understanding of how $L^+$ and $L^1$ spaces are defined (I'm using Folland) is a little hazy. For example, in the Fubini-Tonelli theorem: $\textbf{For the Tonelli part:}$ We start with if ...
0
votes
1answer
43 views

Proposition from the book “Convex analysis and measurable multifunction ”

Please , what is $F_{\sigma}$? And all $U$ can be writen as $\cup F_n$, or only open set ? Thank you
2
votes
0answers
165 views

Exercise on measure theory

I have this exercise: Let $\mathscr {S}$ be a collection of subsets of the set $X$. Show that for each $A$ in $\sigma(\mathscr S)$ there is a countable subfamily $\mathscr C_0$ of $\mathscr S$ ...
3
votes
2answers
110 views

Definition of complete in the context of Lebesgue measurable sets

I came across this statement on Lebesgue measurable sets. The Lebesgue measurable sets are said to be complete because every subset of a null set is again measurable and the lebesgue measurable ...
1
vote
0answers
35 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
5
votes
1answer
265 views

What is “algebra” in $\sigma$-algebra (or “field” in $\sigma$-field)?

I know that $\sigma$ in $\sigma$-algebra stands for the closure under countable union property. What about "algebra"? Surely it cannot algebra over a field or a ring as defined in algebra textbooks. ...
1
vote
2answers
209 views

Measure and Outer Measure Definition

I would like to find exemples to show and demonstrate that each of the statements of the definition of: -measure $\mu\left(\emptyset \right)=0$ $\mu \left( \bigcup A_n\right)=\sum \mu \left( ...
1
vote
1answer
129 views

Meaning of Point Evaluation

I read in some general measure theory books and there is always like "define measure $x$ to be the point evaluation at $y$..." but when I look around online and some other books there is no mention on ...
2
votes
4answers
111 views

What are possible variations of the definition of $\sigma$-additivity?

From what I've read in Wikipedia, $\sigma$-additivity is defined in the following way: Let $\mathcal{A}$ be a $\sigma$-algebra of some underlying set $X$ and let $f$ be a mapping ...
2
votes
0answers
55 views

terminology clarification needed: projection of a measure to an open set

I'm reading in Doobs "Classical Potential Theory and its Probabilistic Counterpart" and I'm having trouble with terminology. Specifically, in Part I chapter 6, he talks about the projection of a ...
6
votes
1answer
874 views

what is f prime?

currently taking Measure and Integration course, which seems to have a different definition of f'. traditionally, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ but in folland's book, it seems ...