Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

learn more… | top users | synonyms (1)

0
votes
2answers
131 views

Proving equality of sigma-algebras

Let $C_1$ and $C_2$ are two collections of subsets of the set $\Omega$. We want to show that if $C_2$ $\subset$ $\sigma$[$C_1$] and $C_1$ $\subset$ $\sigma$[$C_2$], then $\sigma$[$C_1$]=$\sigma$[$C_2$]...
2
votes
1answer
428 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
1
vote
1answer
242 views

Are there open sets of measure zero?

Suppose $A \subseteq \mathbb{R}^n$ is an open set. Can we conclude that $A$ does not have measure zero?? I am trying to find an open set with measure zero, but it seems quite hard to construct one ...
0
votes
1answer
272 views

$f$ being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If $f$ is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have $\...
3
votes
2answers
79 views

lebesgue measurable problem

Let $E$ be a Lebesgue measurable subset of $[0,1]$, and suppose that $m(E)>3/4$. Prove that $(-1/2,1/2) \subseteq E-E$. We use $E-E$ to denote the set $\{x-y:x,y \subseteq E\}$
1
vote
0answers
93 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance $(1-\Delta)^{-...
0
votes
1answer
31 views

For $(a,b)$, if $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) \cap - E)$ then $E$ in $\mathbb{R}$ is measurable

If $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) - E)$ for all open intervals $(a,b)$, then $E$ in $\mathbb{R}$ is measurable. How do I prove this? Totally stuck.
1
vote
1answer
50 views

Weak absolute continuity of measures

I want to show that if we have a function $f \in L^p$ sucht that $||f||_p =1$. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ \exists \delta>...
1
vote
1answer
101 views

In Egorov's theorem, remove the condition $\mu(E) < \infty$ and let the sequence be convergent in measure. The conclusion holds for subsequence

Let $(X,\mathscr{F},\mu)$ be a measure space, $E \in \mathscr{F}$, $\{f_n\}$ is a sequence of measurable functions on $E$, and the sequence converges to function $f$ in measure. Show that $\exists \{...
2
votes
1answer
706 views

Continuous, strictly increasing function that maps a set of positive (lebesgue) measure onto a set of measure zero?

Is there a continuous, strictly increasing (real-valued) function on the interval $[0,1]$ that maps some set of positive (lebesgue) measure onto a set of measure zero? Should I play with cantor ...
0
votes
1answer
79 views

Measure Theory Question 3

Let $E$ be a measurable set with $m(E)<\infty$. Show that there is a descending sequence of open sets $\{G_n\}$ so that $E\subseteq G_n$ for all $n \ \epsilon \ \mathbb N$ and $ \lim_{n\to\infty} m(...
0
votes
3answers
56 views

A question on limit of weak-* convergence of probability measures

Let $(X,\mu)$ be a measure space. Assume $X$ is compact. It is well-known that the space $\mathcal{P}(X)$ of probability measures on $X$ is compact in weak-* topology. Let's consider a sequence of ...
1
vote
1answer
56 views

When is the composition of a function with Dirac delta a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...
1
vote
0answers
90 views

Weak Compactness for measures

Let $\{\mu_k\}_{k>1}$ be a sequence of radon measures on $\mathbb{R}^n$ satisfying $\sup_k\mu_k(K)<\infty$ for each compact set $K\subset \mathbb{R}^n$. Prove that if $\sup_k\mu_k(\mathbb{R}^n)=∞...
5
votes
2answers
267 views

Lebesgue integrability and measurable functions

Let $f$ be a nonnegative function on the reals. What does the (Lebesgue) measurability of $f$ have to do with the (Lebesgue) integrability of $\int f$? I've spent some time studying the definition at ...
1
vote
2answers
613 views

Collection of half open intervals is an algebra, why?

I have to show, why the collection of all finite unions of such half open intervals $(a,b]$ is an algebra and not a sigma algebra. I know that $−∞≤a≤b≤∞$, and have: $$ (a,b)=\bigcup_{n=1}^∞ \left(\...
1
vote
1answer
104 views

Which are the conditions for a Lorentz space $L^{p,q}$ to be o-c?

Which are the conditions for a Lorentz space $L^{p,q}$ to be ord. continuous? ( A Banach function space is o-c $\equiv$ Increasing sequences of order-bounded positive functions converge in norm). ...
4
votes
1answer
59 views

Show that $g\in\mathcal{L}^q(\mu)$.

Let $(X,\mathcal{A},\mu$) be a finite measure space and $p,q\in(0,\infty)$ such that $1/p+1/q=1$. Let $g\in\mathcal{M}(\mathcal{A})$ measurable function such that $$\int |fg|d\mu\leq C\|f\|_p$$ for ...
5
votes
5answers
361 views

real analysis, chebyshev's inequality

Suppose $f$ is a non negative integrable function on a measure space $(X,M,μ)$. Prove that: $$\lim_{t \rightarrow \infty} t\cdot \mu(\{x:f(x)\geq t\} )=0.$$ Can you help me please?
2
votes
1answer
98 views

Differentiability of the composition of maps and Differentiability a.e.

Let $ f: \mathbb{R} \rightarrow \mathbb{R} $ be non constant Lipschitzian function and $g : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable a.e.in $ \mathbb{R} $. Then it is easy to show that ...
2
votes
2answers
269 views

Lebesgue measurable homework problem

Let $X \subseteq \mathbb{R}$. A subset $E \subseteq \mathbb{R}$ is called a hull of $X$ if $E$ is measurable $X \subseteq E$ If $F$ is any measurable set such that $X \subseteq F$, then $E$\ $F$ is ...
3
votes
0answers
81 views

Integration over uncountable set of characters

Let $G$ be a compact (assumed Hausdorff) group and $\hat{G}$ be the set of all characters of irreducible, finite-dimensional representations of $G$. It might occur that $\hat{G}$ is uncountable. It ...
0
votes
1answer
60 views

Scale invariant measures must have power law densities

If $\mu$ is a scale-invariant measure(say on $\mathbb{R}^{+}$), i.e. for any set $A$, $\mu\left(\frac{A}{c}\right)=g\left(c\right)\mu\left(A\right)$ where $c>0$, then is it necessary that $g$ must ...
1
vote
1answer
38 views

$f \le 1 \Rightarrow f =1 $ a.s.

I know the title doesn't say much, but I hope you'll help me nonetheless. Here's my problem. Let $P, Q$ be two probabilistic measures, $P$ is atomless and the measures have the same independent ...
2
votes
1answer
61 views

Lebesgue-Radon-Nikodym Theorem without Hilbert spaces

In my analysis class we are seeing the so called Lebesgue-Radon-Nikodym Theorem. But we prove it the "old fashioned way" without using Hilbert space theory. More precisely, we prove the minimality ...
2
votes
2answers
1k views

Sigma-algebra requirement 3, closed under countable unions.

The requirement for sigma-algebra is that. It contains the empty set. If A is in the sigma-algebra, then the complement of A is there. 3. It is closed under countable unions. My question relates ...
3
votes
1answer
250 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
1
vote
1answer
44 views

If $u \in L^2(\Omega)$, then $\text{sign}u \in L^2(\Omega)$?

If $\Omega$ is a bounded domain and $u$ is in $L^2$, why is $\text{sign}(u) \in L^2?$ I am only stuck with the measurabilituy part. the integral is obviously finite on a bounded domain.
0
votes
1answer
16 views

Continuity at $x$ of increasing function if certain sequences exist

I'm working through the first few chapters of Royden-Fitzpatrick to learn measure theory and I got stuck on this question. Let $f$ be increasing on $I$, an open interval. Then for $x \in I$, $f$ is ...
0
votes
1answer
38 views

Square of absolute value of a function different than square of function

How come if f is measurable, we might have $|f|^2\neq f^2$? Can you provide an example? I think it is true if f is real.
1
vote
1answer
88 views

Girsanov's theorem and simulation of bond prices

Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma dB_t=b(r_t)dt+\...
3
votes
1answer
77 views

Lebesgue integral over “bad” measurable sets

Let $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) be a bounded open domain and $f \in L^\infty(\Omega)$ possibly changes the sign. Assume that the set $$ \Omega^+ := \{x \in \Omega: f(x) > 0 \} $$ has ...
3
votes
1answer
106 views

bounded measure and dense subset of continuous functions

Let $C_0(R^n)$ be the space of continuous functions from $R^n$ to $R$ which vanish at infinity. Let $D$ be a subset of $C_0(R^n)$, I'd like to prove that if D is not dense in $C_0(R^n)$, then there ...
0
votes
2answers
143 views

Motivation behind the proeprties of sigma algebra

What is the motivation behind the class $B$ of all measurable sets to satisfy the following properties : 1) $A_1, A_2 \in B$ implies $A_1 \cup A_2 \in B$ 2) $\{A_n\} \in B $ and $\{A_n\}$ is ...
0
votes
1answer
209 views

Meaure Zero of the XY Plane

Prove that the $XY$-plane has measure zero in $\mathbb R^3$. I am learning about measure zero in Analysis and I understand that it would have measure zero if it can be broken down into a covering ...
3
votes
4answers
2k views

every subset of a measurable set is measurable

Is it true that every subset of a measurable set is measurable? for any measure. So if A is a measurable set then, B as a subset of A must be measurable wrt the same measure.
1
vote
1answer
52 views

Two measures on a same space

I have two measure space $(X, S, \mu_1)$ and $(X, S,\mu_2)$, where $S$ is the minimal $\sigma$-algebra containing sets $T = \{E_i\}_{i \in I}$. Suppose further that $T$ is closed under taking finite ...
2
votes
1answer
224 views

Application of Riesz representation theorem

Suppose the following situation. We have linear functional $l$ on the space $H(\mathbb{C}^n)$ of entire function and wish to find a representation for $l$ with integration against a complex Borel ...
1
vote
0answers
1k views

The Cantor set is nowhere dense

I am considering the so called Cantor ternary set $C$ on $[0,1]$. I have just proved that its Lebesgue measure is $0$. To show that $C$ is nowhere dense, is it correct the following reasoning? By ...
4
votes
2answers
159 views

Intuition behind the failure of unimodularity

If $G$ is a locally compact group then up to normalization it admits a unique Haar measure: a left invariant measure defined on all Borel subsets of $G$, which assigns every compact set a finite ...
0
votes
1answer
38 views

Notation, abbreviation $a.s.$ measure theory

Could you tell me what $m - a. s.$ means in measure theory? Here $m$ is a measure. Thank you.
2
votes
1answer
85 views

Set of simple predictable processes is a vector space

I have a question, which is probably very easy for you to answer. How can I show that the set of simple predictable processes a vector space is? It's clear that I only have to show that the sum of ...
1
vote
1answer
126 views

Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement $$\alpha_k=\inf\{\...
4
votes
1answer
214 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently large....
0
votes
1answer
68 views

Borel-set, open, measurable function.

I have a questions about Borel sets. Here is how they defined in my book: Now they say that, the set consits of open sets. But it must not nececarrily be all open sets on X? The reason this ...
1
vote
2answers
389 views

Must every probability distribution over a countable set be discrete?

Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ...
1
vote
0answers
60 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in L^1_{\mathrm{loc}}(X,\mu),\,w\ge0\,\,...
2
votes
1answer
62 views

Probable use of Radon Measure

The problem is: Suppose $\mu$ is a positive Borel Measure on $\mathbb R^{1}$ which is finite on bounded sets. If $ \forall f,g \in C_{c} ( \mathbb R^{1})$ ; $ \int_{ \mathbb R^{1}} fg d\mu = (\int_{ ...
1
vote
1answer
317 views

weak convergence of a bounded linear operator [duplicate]

I need help with this problem Let $X$ be a reflexive Banach space and $T: X \to X$ a linear operator. Show that $T$ belongs to $\mathcal{L}(X,X)$ if and only if whenever $\{x_n \}$ converges weakly ...
0
votes
1answer
56 views

Projection of a Manifold has measure zero

Let $M$ be a $k$-dimensional sub manifold of $\mathbb{R}^N$, and let $\pi_n:\mathbb{R}^N\to\mathbb{R}^n$ be the canonical projection, with $n>k$. Can we show that $\pi_n(M)\subset \mathbb{R}^n$ has ...