# Tagged Questions

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

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### Injectivity of Fourier transform between $L^1(\mathbb{R})$ and $C_0(\mathbb{R})$

The Fourier transform maps from $L^1(\mathbb{R})$ to $C_0(\mathbb{R})$ where $C_0(\mathbb{R})$ is all continuous functions that vanish as $x \rightarrow \infty$. Now given $f,g \in L^1(\mathbb{R})$, ...
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### Checking Caratheodory-measurability condition on sets of the semiring

Let $\mathcal H$ be a semiring over the set $X$ and $\mu$ a pre-measure defined on $\mathcal H$. Then we associate an outer measure $\mu^\ast$ to $\mu$ (describe here: ...
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### Example for a step function

I need to determine if one can find step functions $f , f_1 , f_2$ such that $f$ is Lebesgue approximate to $f_1 + f_2$ but $f \neq f_1 + f_2$. My justification is that yes we can find such step ...
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### Prove that a polyrectangle in an open set has a superset polyrectangle.

Theorem: Let $S \subseteq \mathbb{R}^n$ be an open set. Let $P \subset S$ be a polyrectangle. Then there exists another polyrectangle $P'$ such that $P \subset P' \subset S$. A ...
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### Can this solution be simplified this way?

I have an exercise which I also have the answer for, but the answer seem very complicated, I am wondering if my simplified solution also is correct, or does it miss something? the exercise: Let ...
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### Total variation of a vector valued measure

If I have understood correctly, a vector valued measure $\mu$ is simply a vector of measures, that is $\mu=(\mu_1,\dots,\mu_n)$, where $\mu_i$ is a possibly signed measure on the measure space ...
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### Mathematical expectation of $F_\xi(\xi)$

Consider $F$ as a distribution function of some random variable $\xi$. The problem I'm trying to solve is to find integral: $$\int_{-\infty}^{+\infty}F(x)dF(x)$$ From what I see, there are two ways ...
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### Vector space of all Lebesgue measurable functions

Let $L_0$ be the vector space of all Lebesgue measurable functions on $[0,1]$ with metric $d(f,g)$ = $\int_{0}^{1} |f(t)-g(t)|/( 1+ |f(t)-g(t)| ) dt$ . Show that $d(f_n,f) \to 0$ iff $f_n \to f$ in ...
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### Creating a monotonous rising function that is not equivalent to a continuous function on any interval

I have the answer given. I do not understand only one step in it. $\mathbb Q$ is countable, so therefore we can put it's points in a array: $\{q_0,q_1,...\}$ and then the function ...
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### Measurable set of real numbers with arbitrarily small periods

I am trying to prove the following exercise (exercise 3, chapter 7 of Rudins Book "Real and Complex Analysis"): Suppose that $E$ is a measurable set of real numbers with arbitrarily small ...
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### Compactness of Set of Probability Measures

I'm currently studying Information Theory from Csiszar and Korner, "Information Theory: Coding theorems for Discrete Memoryless Sources". There are several questions pertaining to the set of all ...
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### $\int f \ d m = \int f c \ d \frac{m}{c}?$

Is it true that if $u = m/c$ is a measure, where $m$ is another measure and $c$ is a positive constant, then $\int cf \ du = \int f \ dm$? If not, what assumptions do we need for it to be true? My ...
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### The collection of all regular sets is a $\sigma$-algebra iff $X$ is regular

Let $X$ be a Hausdorff topological space with finite Borel measure $\mu$. Let $\mathcal{T}$ be the collection of all Borel sets $A$ with $$\mu(A) = \sup\{\mu(C): C \subset A, C \text{ compact}\},$$ ...
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### A Function is indefinitely differentiable

Suppose $f(x)=e^{-\frac{1}{x-a}-\frac{1}{x-b}}$ if $x \in (a,b)$, and $f(x)=0$ otherwise, show that $f \in C_{c}^{\infty} (R)$ I think I have to just find the general form of the $n^{th}$ ...
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### Comparing the size of two measure and their extensions.

Let $X$ be a non-empty set and $\tilde{H} \subseteq P(X)$ be a semiring on $X$. Furthermore, let $\mu$ and $v$ be measures on $\tilde{A}$, where $\tilde{A}$ is the $\sigma$-algebra generated by ...
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### Extension of a measure by using a measurable hull of a non-measurable set.

Let $(X,\tilde{A},\mu)$ be a finite measure space and let $D \subseteq X$, but $D \notin \tilde{A}$. Furthermore, let $M\in \tilde{A}$ be a measurable hull of $D$. This means that $D \subseteq M$ and ...
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### Proving that a function is L1

Suppose $f \in L^1([0,b])$ and $g(x)=\int_x^b{\frac{f(t)}{t}dt}$ , prove that $g\in L^1([0,b])$ and $\int_{0}^{b} g(x) dx = \int_{0}^{b} f(t) dt$. Assume we are not allowed to use integration by ...
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### Question about total variation, positive variation and negative variation

I have a question about the following problem: Let $f:[a,b] \to \mathbb{R}$ be a function of bounded variation with $f(a) = 0$ and $f_1$, $f_2$ be two increasing function such that $f_1(a) = 0$ ...
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### An intuitive interpretation of stopping time

I have the following definition of exercise time. Let $T\in\mathbb{N}$ with $T>0$, let $(\Omega,\mathcal{F})$ be a probability space with the $\sigma$-algebra $\mathcal{F}=2^{\Omega}$ and let ...
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### If $f \in L^{p_1}$ and $f \in L^{p_2}$ with $1 \le p_1 \lt p_2 \lt \infty$, then $f \in L^{p}$ for all $p$ such that $p_1 \leq p \leq p_2$.

Let $(X, \Sigma, \mu)$ be a measure space. If $f \in L^{p_1}$ and $f \in L^{p_2}$ with $1 \le p_1 \lt p_2 \lt \infty$, then $f \in L^{p}$ for all $p$ such that $p_1 \leq p \leq p_2$. My attempt Let ...
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### How is an adjunction map of a probability space well-defined?

The book says something like: Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $\Delta \notin \mathcal{F}$, suppose that for all $F \in \mathcal{F}$ such that $F \supset \Delta$, we have ...
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### $f$ locally bounded, nonnegative, and measurable function integrable iff series $\int_{n=1}^{\infty}a_{n}$ converges absolutely
Suppose $f$ is a locally bounded, nonnegative, and measurable function on $[1,\infty)$ and define $\displaystyle \int_{n}^{n+1}f$, $\,\,\forall n \in \mathbb{N}$. Then, is it true that $f$ is ...
### $\sigma$-finite measure on $\mathbb{R}$ that maps half-open intervals to $\infty$
Consider $\mathbb{R}$ equipped with the Borel-$\sigma$-algebra $B$ and a measure $\mu : B \rightarrow [0, \infty]$. The measure $\mu$ is called $\sigma$-finite, if there is a sequence $A_1,A_2,...$ of ...