# Tagged Questions

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

81 views

### Does every Lebesgue measurable set $A$ with $m(A)>0$ contain at least an open subset?

Does every Lebesgue measurable set $A$ with $m(A)>0$ contain at least a non-empty open subset? I came across this question when I was reading the Stein and Shakarchi's Real Analysis book. Is ...
68 views

### Compute the limit of $\int_\mathbb{R} f(x)\sin (nx)$ when $n\to\infty$, for $f \in L^1$

Let $f \in L^1(\mathbb{R})$. Find $$\lim_{n \rightarrow \infty} \int_{-\infty}^\infty f(x)\sin(nx) dx \,.$$ LDCT is a no go, as well as MCT and FL, which are really the only integration ...
25 views

### For a 2-dimensional random vector, express the probability of being in a rectangle in terms of CDF

For $a_1\le b_1$ and $a_2\le b_2$, show that $$P\{ a_1<X_1\le b_1,a_2<X_2\le b_2 \}= F(a_1,a_2)+F(b_1,b_2)-F(b_1,a_2)-F(a_1,b_2)$$ Here $F(a,b)=P\{X_1\le a,\ X_2\le b\}$, the cumulative ...
20 views

### Proving that x $\in \overline{\lim}(A_{n}$ implies for each N there exists at least one n$\geq$N

Letting {A$_{n}$} be a sequence of sets. We are given the definition $$\overline{\lim}(A_{n})= \cap_{n\geq 1}\cup_{k\geq n}A_{k}$$ The question wants us to prove that x $\in\overline{\lim}(A_{n})$ ...
59 views

### Uniformly integrable implies integrable?

The term "uniformly integrable" sounds (to a layman like me) to be stronger than integrable. Just like how uniformly convergent is stronger than simply being convergent. However, from the definition ...
35 views

### Does there exist a sigma-algebra $F$ such that $f$ is $F/\mathbb{B}$ measurable if and only if $f$ is a constant function?

Let $f$ be a function from $(\mathbb{R},F) \rightarrow (\mathbb{R},\mathbb{B})$ where $\mathbb{B}$ denotes the borel sigma algebra. Does there exist a sigma-algebra $F$ such that $f$ is ...
71 views

### A confusing question in Royden's Real Analysis

Let $I$ be a closed and bounded interval, and $E$ a measurable subset of $I$. Let $\epsilon>0$. Show that there is a step function $h$ on $I$ and a measurable subset $F$ of $I$ for which ...
43 views

### Definition of essential maximum and essential supremum

Let $\Omega \subset R^n$ a bounded domain. Consider $u \in L^1(\Omega)$. The definition of essential supremum is well known. I am reading the book "Linear and quasilinear elliptic equations" of ...
32 views

### a countably additive function which is not a measure

Is there a simple example showing that a non-negative countably additive function on a $\sigma$-algebra fails to be a measure?
31 views

23 views

### Heuristic statement of the monotone class lemma

From Folland's Real Analysis: Modern Techniques and Their Applications we have: The Monotone Class Lemma (MCL): If $\mathscr{A}$ is an algebra of subsets of $X$, then the monotone class ...
75 views

### Characterization of smallest $\sigma$-Algebra on $\Omega=[0,1)$ that contains disjoint intervals of the form $[a,b)$ where $0<a<b<1$

Problem: Let $\Omega$=[0,1) and $0=a_0 < a_1< \dots < a_n =1$ where $n \in \mathbb{N}$ Let $X:= \lbrace [a_{i-1}, a_i) : i =1,2, \dots , n \rbrace$ and find an identity for $\sigma(X)$ ...
28 views

97 views

### What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
41 views

86 views

79 views

### Is total variation a continuous map from complex measures to positive measures?

The following question arises naturally in my current research. It seems to be a basic problem in measure theory, and therefore I guess that answers to it can be found in some textbooks. However, I ...
55 views

### Strong Sobolev inequality

Take $B(0,1)$ the ball in $\mathbb{R}^2$ with the normalized Lebesgue measure $\lambda$ such that $\int_{B(0,1)} d \lambda=1.$ Now, I want to show, or give a counterexample that this is false, that ...
61 views

### Measurable function in $L^2$-norm

Let $f$ be a measurable function. Assume you know that $$\sup_{||g||_2=1} \left|\int fg\,\right|$$ exists. Does this mean that $$\sup_{||g||_2=1} \left|\int fg\,\right| = ||f||_2?$$
244 views

### Product of metric outer measures

The problem below has been asked recently already but, as a naive user, I got burned (well singed perhaps) because I asked the question in the wrong place. So if this looks like a redundant question ...
86 views

### A function $f$ that is not in any $L^p$ but the measure of $\{|f|>t\}$ is bounded by $C/t$

How can I find a function $f$ such that $f \notin L^{p} (\mathbb{R})$ for all $p$ but you can find a constant $c>0$ for it with $m(x \in \mathbb{R} \, s.t. |f(x)|>t) \leq \frac{c}{t}$ for ...
Let $f,g : \mathbb{R}^{n}\to [0,+ \infty]$ 2 Borel measurable functions. Show that {${x \in \mathbb{R}^{n}| f(x)<g(x) }$} is a Borel set. I don't know how to start with the problem. At the ...