Articles of lebesgue measure

theorem of regularity of Lebesgue measurable set

I was reading a proof regarding the condition for Lesbesgue measurable set. Specifically it is the Theorem 2.24 and the proof of the theorem here: https://www.math.ucdavis.edu/~hunter/m206/measure_notes.pdf In the theorem, it says a set A is lebesgue measurable if and only if there is an open set G where $A\subset G$ such that $\mu^{*}(G\setminus A) \le […]

Show that $C_1= [\frac{k}{2^n},\frac{k+1}{2^n})$ generates the Borel σ-algebra on R.

Let $C_1$ be the collection of intervals of the form $[\frac{k}{2^n},\frac{k+1}{2^n})$ where $n = 1, 2, 3, . . . and $ k ∈ Z , together with the empty set. a) Show that C_1 generates the Borel σ-algebra on R. b) Show clearly and explicitly that $C_1$ is a p-system. I think for first […]

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k \hookrightarrow \mathbb{R}^n$ we have $\mu_{k}(U \cap i(\mathbb{R^k}))=0$. Does this imply $\mu_n(U)=0$?

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\frac1n\sum\limits_{i=1}^n f(a_i) \to \lambda(f):=\int_0^1 f(x)\,dx$ for all continuous $f:[0,1]\to \mathbb{R}_{\ge 0}$. In the proof I used the fact that if $T_g(x)=(g+x) \pmod 1$, then $\lambda(f\circ T_g)=\lambda(f)$ for all continuous […]

Mollifiers: Approximation

Problem Given a mollifier: $\varphi\in\mathcal{L}(\mathbb{R})$ Then it acts as an approximate identity: $$f\in\mathcal{C}(\mathbb{R}):\quad\int_{-\infty}^\infty n\varphi(nx)f(x)dx\to f(0)\cdot\int_{-\infty}^\infty\varphi(x)dx$$ How to prove this under reasonable assumptions? Example As an example regard the Gaussian: $$f\in\mathcal{C}(\mathbb{R}):\quad\frac{n}{\sqrt{\pi}}\int_{-\infty}^\infty e^{-(nx)^2}f(x)\mathrm{d}x\to f(0)$$ (This is a useful technique when studying operator semigroups.)

Lebesgue integral on any open set is $\ge 0$, is it still $\geq 0$ on any $G_{\delta}$ set?

Definition of Lebesgue measurable function: Given a function $f: D \to \mathbb R \cup \{+\infty, -\infty\}$, defined on some domain $D \subset \mathbb{R}^n$, we say that $f$ is Lebesgue measurable if $D$ is measurable and if, for each $a\in[-\infty, +\infty]$, the set $\{x\in D \mid f(x) > a\}$ is measurable. If $f$ is an extended […]

example of maximal operator that is integrable

We know that there are no nonzero functions $f \in L^1(\mathbb R^n)$ such that $Mf \in L^1(\mathbb R^n)$, where $Mf$ is the Hardy Littlewood maximal function. Can we find a maximal operator that is integrable for nonzero functions? More precisely, I would really appreciate some help with the following: Let $ \phi \in C^\alpha (\mathbb […]

$f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable

I know there are similar questions up proving this, but I had a question specific to the following proof (specifically in bold): Let $f$ be a Lebesgue measurable function on $\mathbb{R^n}$. Then the function $f(x)$ considered as a function of $(x,x)\in \mathbb{R^n}$ is Lebesgue measurable. The linear transformation given by $T:(x,y)\rightarrow (x−y, y)$ is invertible, […]

$L^1$ is complete in its metric

Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as follows. First assume a Cauchy sequence $(f_n)\in L^1$, then we try to extract a subsequence $\left(f_{n_k}\right)$ of $(f_n)$ which converges […]

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family $\{E_{i}\}$ of pairwise disjoint Borel sets of $\mathbb R$), and for which $\mu(E)$ is finite if the closure of $E$ is […]