# Tagged Questions

**2**

votes

**1**answer

180 views

### Measure of the same set in different models of ZF

Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models.
Is ...

**2**

votes

**2**answers

132 views

### Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...

**3**

votes

**0**answers

64 views

### Almost everywhere in a rectangle [duplicate]

I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.
Let $S$ be a subset of a region, say $R$ which is enclosed by a ...

**-4**

votes

**1**answer

148 views

### Is there always a subspace of $L^p$ isomorphic to direct sums of $\ell^2,\ell^p$? [closed]

It is known that each $L^p$ (on a space with finite measure like $[0,1]$) $p>1$ space contains an isomorphic complemented copy of $\ell^2$ and $\ell^p$. I think this is the Kadets-Pelczynski ...

**3**

votes

**1**answer

124 views

### Multivariate monotonic function

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...

**1**

vote

**0**answers

102 views

### Relationship between weak Lp and strong Lq topologies for q<p

Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If ...

**8**

votes

**1**answer

253 views

### Can there be a measurable set that integrals have the same given value if their integral on $\mathbb{R}$ are the same?

We know for an integrable function $f$, if $\int_\mathbb{R} f=1$, then $\forall \lambda\in [0,1] $, there exists a measurable set $E$ that $\int_E f=\lambda$.
Now consider integrable functions $f$ ...

**2**

votes

**1**answer

257 views

### Operation on measurable sets in lines, containing an interval?

Question 1: In $\mathbb{R}^2$, let $l_1$,$l_2$ be two parallel lines and $l_3$ another line which is not parallel to $l_1$. Given two measurable sets $E_1$ and $E_2$ in $l_1$ and $l_2$ respectively, ...