1
vote
0answers
24 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
1
vote
1answer
47 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
1
vote
1answer
22 views

Is a set of jointly bounded functions over a compact domain compact under p-norm?

Let $X$ be a metric space and a measurable space. Let $K$ be a compact set of nonzero measure and $r> 0$. Is a set $\{ f: K\rightarrow \mathbb R| |f|\leq r$ almost everywhere$\}$ compact with ...
1
vote
2answers
33 views

Space of probability measures total bounded?

I want to consider a space of probability measures on some set $\Omega$. It's complete (am I right?). But I don't know whether it's total bounded. Actually, I want to prove that the space of ...
2
votes
0answers
69 views

Measures with bounded total variation norm compact in $M(X)$?

Let $X$ be a separable, metric, compact space. (e.g. an interval in $\mathbb{R}$ like $[0,10]$). Let $M(X)$ be the set of all finite signed measures over $X$ with weak-*-topology (in probability ...
5
votes
2answers
69 views

On the compacity of the space of probability measures

Let $X$ be a complete metric space, and denote by $\mathcal{P}(X)$ the set of probability measures on $X$. I am interested in proving that if $X$ is compact then $ \mathcal{P}(X)$ must be compact in ...
0
votes
2answers
27 views

Compactness of a set in a Locally compact Hausdorff space

My question is about the following lemma: Let $X$ be a locally compact Hausdorff space, and $V$ be a nbhd. of a point $x \in X$. Then there is a nbhd. $U_{x}$ of $x$ such that $\overline{U_{x}}$ is a ...
3
votes
0answers
99 views

Prokhorov theorem in locally compact Hausdorff space?

Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general ...
0
votes
1answer
166 views

Compact subset of a measurable set

Let S contained in R^n, be a measurable set with μ(S) < ∞, and let ε > 0 and also $$\epsilon<\mu(S)$$be a positive real number. Show that there exists a compact subset K contained in S such ...
0
votes
1answer
36 views

Relative compactness in $L^2(0,T,BD(\Omega))$

Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary and I have a sequence of functions $(v_n)_n$ such that $$\forall n ...
1
vote
1answer
266 views

Is there a compact set which is not Jordan measurable?

Is there a compact set which is not Jordan measurable? Intuitively, the answer seems like there is no such set but I could not find a proof anywhere. Does anyone know of a proof or a counter example?
3
votes
1answer
55 views

Question on compactness of $\Delta(X)$ with the weak$^{\ast}$ topology for compact $X$

Let $X$ be a metric space. $\Delta(X)$ is the set of all probability Borel measures on $X$(which is regular, of course).$\Delta(X)$ is endowed with weak$^{\ast}$ topology for which the mapping $\mu ...
2
votes
0answers
175 views

Every Borel set is the union of an increasing sequence of Bounded Borel sets?

I am currently working with the book by Halmos, and i can't quite get past this one. It states that: "Every Borel set can be written as an increasing sequence of Bounded Borel sets" In this case $X$ ...
0
votes
1answer
124 views

Compactness and compact-finite measure in Lusin theorem (Rudin)

I have two questions about some hypotheses in Lusin's theorem as stated in Rudin's "Real and Complex Analysis". The proof initially deals with a subcase, that is the function $f$ is supposed to be ...
2
votes
2answers
166 views

Compact inclusion in $L^p$

Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$? What is the counterexample if what I said is wrong? Thank you.
32
votes
4answers
642 views

To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
5
votes
1answer
148 views

Proving $\mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$

Can someone please help me show, why in a compact metric space $(X,d)$ we have have $$ \mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$$ and $$ \mu(A)=\sup\{\mu(K) \mid K\subseteq A, K \text{ ...
4
votes
0answers
128 views

Constructing the support of a Borel measure

From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition. Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous ...
5
votes
1answer
107 views

Compactness of Convolutions of Compact Measures

This regards measures on $d$-dimensional Euclidean space $\mathbb R^d$ and their associated densities. A super-level set of a density $f : \mathbb R^d \to \mathbb R^+$ at level $t$ is the set $\{x \in ...
3
votes
2answers
253 views

Support of regular Borel Measure

This question is elementary and hence might be a duplicate. From Rudin, Real and Complex Analysis, page 57. Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$: assume $\mu(X)=1$. ...
3
votes
2answers
198 views

$X$ is locally compact Hausdorff space, $\mu$ is Borel regular measure. How to prove $\mu$ is cover $[0,\mu(A)]$

I mean $X$ is locally compact Hausdorff space, $\mu$ is Borel regular measure, and $\mu(\{x\})=0$. For any subset $A$ with finite measure. How to prove for any $0<b<\mu(A)$, we always can find a ...
1
vote
2answers
112 views

Is it true that a set is compact iff it is closed, bounded, and has finite measure?

I'm sure that this holds for $\mathbb{R}^n$ and for $L^p$ spaces. Is it true in general?
0
votes
3answers
914 views

is the smallest $\sigma$-algebra containing all compact sets the Borel $\sigma$-algebra

Let $R$ be the smallest $\sigma$-algebra containing all compact sets in $\mathbb R^n$. I know that based on definition the minimal $\sigma$-algebra containing the closed (or open) sets is the Borel ...
8
votes
1answer
302 views

Topology on the set of partitions

Let $X$ be the set of all partitions of $[0,1]$ such that each element of the partition is Lebesgue-measurable. Let $Y$ be the set of all partitions of $[0,1]$ such that each element of the ...