All Questions
Tagged with measure-theory integration
2,732
questions
243
votes
1
answer
14k
views
Is this really a categorical approach to integration?
Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A ...
- 45.7k
77
votes
2
answers
9k
views
Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus):
the indefinite ...
user9464
71
votes
2
answers
29k
views
Integration with respect to counting measure.
I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let $f:\mathbb{N}\rightarrow{\mathbb{R}...
user54992
66
votes
4
answers
8k
views
Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
- 2,231
58
votes
5
answers
28k
views
Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure
Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure? Specifically, most references that I can find give a change of variables ...
- 581
50
votes
8
answers
6k
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 ...
- 2,924
36
votes
3
answers
12k
views
If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?
Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ ...
- 11.8k
34
votes
4
answers
5k
views
Is there any difference between the notations $\int f(x)d\mu(x)$ and $\int f(x) \mu(dx)$?
Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$$\int f(x)d\mu(x)$$
and the notation
$$\int f(x) \mu(dx)$$?
- 519
32
votes
2
answers
13k
views
Integration with respect to a measure
I am trying to get an explanation in words, or math, of what the $d\mu$ means in an integration statement. Such as:
$$\int f \ d\mu$$
How does the measure change our old "calculus" notion of ...
- 539
30
votes
2
answers
17k
views
Understanding the assumptions in the Reverse Fatou's Lemma
Fatou's Lemma says the following:
If $(f_n)$ is a sequence of extended real-valued, nonnegative, measurable functions defined on a measure space $\left(\mathbf{X},\mathcal{X},\mu\right)$, then
$$
...
- 1,732
30
votes
1
answer
964
views
We have sums, series and integrals. What's next?
We know how to sum or average a finite number of terms: sums.
We know how to sum a countable infinite number ${\beth_0}$ of terms: series.
We know how to sum ${\beth_1}$ terms: integrals.
How to ...
- 377
28
votes
1
answer
2k
views
Reinventing The Wheel - Part 2: The Lebesgue Integral
Disclaimer
After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
- 16.5k
27
votes
6
answers
33k
views
Meaning of measure zero
My book describes measure zero as following:
A set of points on the $x$-axis is said to have measure zero if the sum of the lengths of intervals enclosing all the points can be made arbitrarily ...
- 970
27
votes
2
answers
15k
views
When can the order of limit and integral be exchanged?
I was wondering for a real-valued function with two real variables, if there are some theorems/conclusions that can be used to decide the exchangeability of the order of
taking limit wrt one ...
- 47.7k
27
votes
1
answer
6k
views
Sum of measures and integral
Suppose we have a measure that can be expressed as linear combination of 2 measures: $m=am_1+bm_2$. What does this imply for the calculation of the integral? Do we have:
$$\int f\,dm=a\int f\,dm_1 + ...
- 1,125