All Questions
28
questions
0
votes
0
answers
80
views
Does the Staircase Paradox not contradict the Monotone Convergence Theorem?
I recently came across the Staircase Paradox (or Rising Sea Level Paradox) and have been trying to reconcile it with the Monotone Convergence Theorem (MCT) in Measure Theory. The Staircase Paradox is ...
- 1,459
1
vote
1
answer
859
views
Continuous function on compact is integrable
For a compact subset $X \subset \mathbb{R}^n$ and a continuous function $f : X \to \mathbb{C}$, $f$ is integrable on $X$ ( as both $X$ and $f$ will be bounded).
Does this generalise? That is, if $X$ ...
- 105
2
votes
1
answer
43
views
Prove that a Collection of subsets of $E$ is a $\sigma$-algebra [closed]
Prove that:$\;$ $\mathscr C$ is a $\sigma$-algebra,with$\;$$E$ $\;$ is a infinite set.
$$\mathscr C\equiv\{A\subset\mathbb E\,|\,A\text{ is countable or }A^c\text{ is countable}\}.$$
2
votes
0
answers
151
views
Image of a measure zero set is zero
Prove that if $\varphi:\Bbb R^n\to\Bbb R^n$ is a homeomorphism, then there is a set $Z\subset\Bbb R^n$
of measure zero $|Z|=0$ such that for each measurable set
$A$ the following implication is ...
- 21
2
votes
1
answer
102
views
Is it possible to find non constant, continuous simple function?
Is it possible to find non constant, continuous simple function ? While calculating Lebesgue integral of $f(x) =x^{2}$ on $ \mathbb R $.
We have constructed simple function which is basically function ...
3
votes
0
answers
303
views
Relationship between Product of $L^p$ spaces an product measures
Let $(X,\Sigma,\mu)$ and $(Y,\Gamma,\nu)$ be $\sigma$-finite measure spaces. Is there a relationship between $L^p_{\mu}(\Sigma)\times L^p_{\nu}(\Gamma)$, equipped with the product of their $L^p$-...
- 6,779
2
votes
0
answers
163
views
If $S\subseteq\Bbb R^n$ is rectifiable then its projection $S'$ in $\Bbb R^{n-1}\times\{t\}$ is rectifiable too.
Definition
Let $A$ be a subset of $\Bbb R^n$. We say $A$ has measure zero in $\Bbb R^n$ if for every $\epsilon>0$ there is a covering $Q_1,Q_2,...$ of $A$ by countably many rectangles such that
$$
\...
- 4,906
1
vote
2
answers
125
views
Certain open subsets of $L^1$ for $\sigma$-finite measure
Let $\mu$ be a $\sigma$-finite Borel measure on a metric space $X$, let $B$ be a Borel subset of $X$ of positive $\mu$-measure. Then when does the set
$$
\left\{
I_K g:\, g \in L^1_{\mu}(X)
\right\}\...
- 6,779
0
votes
1
answer
163
views
If $R\subseteq\Bbb{R}^n$ is a rectangle then $m(R)=0\Leftrightarrow v(R)=0\Leftrightarrow\overset{°}R=\varnothing$.
What shown below is a reference from "Analysis on manifolds" by James R. Munkres
Definition
Let $A$ a subset of $\Bbb{R}^n$. We say $A$ has measure zero in $\Bbb{R}^n$ if for every $\...
- 4,906
0
votes
1
answer
38
views
Factorization of Characteristic Function
Let $X$ be an open subset of $\mathbb{R}^n$ and for any subset $A\subseteq \mathbb{R}^n$, let $\chi_A$ be the characteristic function of $X$, i.e.:
$$
\chi_X(x):= \begin{cases}
1 & : x \in A\\
0 &...
- 6,779
1
vote
1
answer
40
views
$M \subseteq \mathbb R$ exist, so that $\lambda^{1}(M \cap I)= \frac{1}{2}\lambda^{1}(I)$ for any bounded Interval $I\subseteq \mathbb R$. [duplicate]
Does a Lebesgue-measurable Subset $M \subseteq \mathbb R$ exist, so that $\lambda^{1}(M \cap I)= \frac{1}{2}\lambda^{1}(I)$ for any bounded Interval $I\subseteq \mathbb R$.
My idea:
Assume that ...
- 998
4
votes
2
answers
58
views
Show that for $f,g \in L^{1}(G)$ we have $f*g \in L^{1}(G)$.
Let G be a locally compact group and consider ($L^{1}(G), \|\cdot\|_{1}$) where $\|f\|_{1}=\int_{G}|f(x)|dx$ for $f \in L^{1}(G)$.
Define convolution $*$ by $(f*g)(y)=\int_{G}f(x)g(x^{-1}y)dx$ for $f,...
- 461
1
vote
1
answer
97
views
Equality of Radon measure
Let $X$ a Hausdorff locally compact topological space. Let $\mu$ and $\nu$ two Radon measures on $X$.
The following result is true ? Why ?
if $$ \forall \varphi \in C^{0}(X), \int{\varphi d\mu} = \...
user347910
1
vote
0
answers
32
views
What does it mean for a map taking a measure to an integral to be continuous?
Suppose $\mathcal{M}$ is a set of probability measures. What does it mean to say that a mapping $\mu \rightarrow \int_{\mathbb{R}}f \ d\mu$ is continuous on $\mathcal{M}$? How could one prove that ...
- 944
2
votes
0
answers
51
views
Under which assumptions are two Borel measures equal?
Assume $(X,\tau)$ is a topological space and $\mathcal B (X)$ the corresponding Borel $\sigma$-Algebra. Let $\mu$ and $\nu$ be two Borel measures on $\mathcal B (X)$. Now let $Q$ be some set of ...
- 1,305