Tagged Questions

For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

36 views

How can I find the measure of $B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$? [closed]

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
94 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
38 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
64 views

Is it correct to interpret the “dx” in the standard notation for integrals as the Lebesgue measure?

Ok so I am in my Calc I class for the summer and we are just beginning to talk about integrals. I know a little bit about measure theory and the Lebesgue integral and why is it more general than the ...
45 views

Lebesgue integral of $\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}$ on an infinite cylinder

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell, and let $\boldsymbol{r}\in\mathbb{R}^3\setminus\overline{V}$ be any point external to $V$. I intuitively suppose that ...
28 views

On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
34 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
30 views

51 views

When we can permute between the integral and convex hull?

Is there a relation between the following expressions? $$\operatorname{conv}\left(\int_{0}^{t} f(s,x)ds :x \in A \right)$$ and $$\int_{0}^{t} \operatorname{conv}(f(s,x):x \in A)\ ds$$ where $A$ ...
43 views

A $\sigma$-finite Borel measure $\mu$ on $\mathbb R$ with $\mu(\mathbb R)=\infty$ s.t. $\int f \,d\mu=\int g \,d\mu \implies f=g$ pointwise?
This is the final part of a problem on an old Analysis preliminary exam at my institution. We are given that $f,g\in L^1(\mathbb R,\mu)$ and $f\leq g$. For earlier parts of the problem, I've already ...