All Questions
67
questions
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votes
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34
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Measure Theory: Proof regarding a measurability of a function [closed]
Is my proof for the following problem correct?
Let $f:X\rightarrow[0,\infty)$ and $X=\bigcup_{i\in\mathbb{N}}A_i$ be measurable.
Prove: $f$ is measurable $\iff$ $f\cdot\chi_{A_i}$ is measurable.
Proof:...
user1197316
5
votes
1
answer
405
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How to show that two integrals are equal?
In my advanced calculus course, I am struggling with the following problem where we need to show that the two integrals are equal.
Consider a function $g:[0,1] \times [0,1] \to \mathbb{R}$
defined by
$...
- 159
0
votes
1
answer
168
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Calculating a Volume Integral with the Divergence Theorem
Hello I have been trying to apply the Divergence Theorem to the following problem but i seem to either missinterpert or not understand the problem.
The following integral should be calculated with the ...
user1197316
0
votes
0
answers
52
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Problem 3-26 in "Calculus on Manifolds" by Michael Spivak. How to solve this problem using Fubini's theorem effectively?
The following problem is Problem 3-26 in "Calculus on Manifolds" by Michael Spivak.
I could solve this problem directly from the definition of "integrable" and "Jordan-...
- 1,138
1
vote
0
answers
149
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How to use the hint by the author? (Problem 3-23 in "Calculus on Manifolds" by Michael Spivak.)
Problem 3-23.
Let $C\subset A\times B$ be a set of content $0$. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B:(x,y)\in C\}$ is not of content $0$. Show that $A'$ is a set of measure ...
- 1,138
0
votes
0
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57
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The definition of integrals over sets which are not rectangles. ("Calculus on Manifolds" by Michael Spivak.)
In "Calculus on Manifolds" by Michael Spivak, the author defined integrals over sets which are not rectangles as follows.
We have thus far dealt with the integrals of functions over ...
- 1,138
1
vote
0
answers
179
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Problem 3-21 in "Calculus on Manifolds" by Michael Spivak.
The following problem is Problem 3-21 in "Calculus on Manifolds" by Michael Spivak.
I solved this problem but I am not sure if my solution is right. Is my solution right? If so, how to ...
- 1,138
0
votes
1
answer
76
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A question about the Integral of an odd function with respect to a finite measure.
Let $\mu$ be a measure defined on borelians of $\mathbb R^2$ such that $\mu(\mathbb R^2)< \infty$. Let $R = [-r,r]\times [-r,r]$, with $r>0$, and $Q = \mathbb R^2 \setminus R$.
How to show that:
...
- 75
0
votes
0
answers
76
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Variable change in functional integral
Suppose I have following functional integral
$$ \int \mathcal{D}x\ f(x) \tag{1}$$
where $$\mathcal{D}x=\prod_t dx(t) \tag{2}$$
is the functional measure.
Now, I want to expand $x(t)$ as
$$x(t)=\sum_n ...
- 225
1
vote
0
answers
50
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$A$ is $J$-measurable, $A$ is null $\implies\text{vol}(A) = 0$
I'm trying to solve the following problem,
Let C $\subset \mathbb{R}^n$ a J-measurable set such that C has zero measure. Show that,
vol(A) = 0.
I only have real analysis tools and I try a ...
- 43
5
votes
2
answers
242
views
Criterion for local integrability of $1/f$ for $f$ a smooth function from $\mathbb R^2$ to $\mathbb C$
Suppose I have a function $f=u+iv$, $f:\mathbb{R}^2\to \mathbb{C}$ which is smooth, in the sense that $u$ and $v$ are smooth. I am wondering what some criteria are for
$$
\int_{B_r(x_0,y_0)}\frac1 f\...
2
votes
0
answers
53
views
understanding surface element in $n \geq 4$ dimensions in proof of Mean value property
I took a basic multivariable calculus class as an undergrad where I saw Green's theorem, Stokes theorem, and Divergence theorem without proof. I know how to use and make sense of them in dimension $n \...
- 620
2
votes
0
answers
59
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$\int_W x^2y^2 = \int_V x^2y^2$. Is my proof of this equation right? ("Analysis on Manifolds" by James R. Munkres)
I am reading "Analysis on Manifolds" by James R. Munkres.
The following example is EXAMPLE 4 on p.149:
EXAMPLE 4. Suppose we wish to integrate the same function $x^2y^2$ over the open set $$...
- 8,740
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votes
0
answers
49
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Is there $(f,S)\in X$ such that $\{x\in S\mid f(x)\neq 0\}$ does not have measure zero? ("Analysis on Manifolds" by James R. Munkres)
I am reading "Analysis on Manifolds" by James R. Munkres.
Theorem 13.5(on p.109)
Let $S$ be a bounded set in $\mathbb{R}^n$; let $f:S\to\mathbb{R}$ be a bounded continuous function. Let $E$ ...
- 8,740
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votes
0
answers
52
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About Theorem 13.6 on p.110 in "Analysis on Manifolds" by James R. Munkres. An integral over the empty set.
I am reading "Analysis on Manifolds" by James R. Munkres.
Theorem 13.6. Let $S$ be a bounded set in $\mathbb{R}^n$; let $f:S\to\mathbb{R}$ be a bounded continuous function; let $A=\...
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