Articles of measure theory

Help in a problem about Lebesgue integration inequality

Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$ Show that: $$f \in L_1\iff\sum_{n=1}^{\infty}n\mu(E_n)<+\infty$$ More generally for $1 \le p <\infty$ show that: $$f\in L_p\iff\sum_{n=1}^{\infty}n^p\mu(E_n)<+\infty$$ Here is what I got by reading an intempt to prove the statement: $$\chi_{E_1} + \frac{1}{2} \sum_{n=2}^{\infty}n\chi_{E_n}\le \sum_{n=1}^{\infty}(n-1)\chi_{E_n}\le|f|\le […]

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that $\Phi'(u)\in W_0^{1,2}(\Omega)$. Is it true that $\Delta (\Phi\circ u)$ is a measure and $$\Delta (\Phi\circ u)=(\Phi”\circ u)|\nabla u|^2+(\Phi’\circ u)\Delta u, \tag{1}$$ If the equality $$\int_\Omega \phi […]

Equivalent Definition of Weak $L^{p}$ (Quasi-) Norm

For a sigma-finite measure space $(X,\Sigma,\mu)$, the weak $L^p$ (hereafter denoted $L^{p,\infty}$) is defined by $$\|f\|_{L^{p,\infty}}:=\sup_{t>0}t\mu(|f|>t)^{1/p}, \qquad (1\leq p<\infty) \tag{1}$$ One can also show that this equals the infimum over all constants $C>0$ such that $\mu(|f|>t)\leq C^p/t^p$ for all $t>0$. Let $1\leq p_{0}<p_{1}<\infty$ be fixed, and let $p=(t/p_{0})+(1-t)/p_{1}$ for $t\in (0,1)$. In an exercise in […]

A measure which is not continuous from above

Let $\Omega= \mathbb{N}$, $F = P(\Omega)$, and $A_n = \{j \mid j \in\mathbb{N}, j \geq n\}$, $n \in\mathbb{N}$. Let $\mu$ be the counting measure on $(\Omega,F)$, so that $\mu(A) = |A|$. I need to show that $$\lim_{n\to\infty} μ(A_n) \neq \mu\bigg(\bigcap_{n\geq 1} A_n\bigg).$$ Now, for a fixed $n$, $μ(A_n)$ cannot be finite, because it is equal […]

Series of integrable functions converges pointwise almost everywhere

I need some help, solving the following problem I found in my textbook. QUESTIONS APPEAR IN BOLD CAPITALS. Let $(X,\Sigma,\mu)$ be a measure space and $f_n \colon X \to \mathbb{C}$ ($n \in \mathbb{N}$) be a sequence of integrable functions such that $$\sum_{n=1}^\infty \int_X |f_n| \mathrm{d}\mu < \infty.$$ Then $\sum_{n=1}^\infty f_n$ converges almost everywhere to an […]

Left and right derivatives of characteristic function $X_Q$

Find the left and right derivatives of a characteristic function of $Q$. My attempt: I tried deriving the result from the definition of right and left derivatives, which are $$D^+f(x)=\lim\limits_{h \to 0}\left[\sup\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right] \text{ and } D^-f(x)=\lim\limits_{h \to 0}\left[\inf\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right]$$ Now, let $Q$ be a set of rationals where $f(x) = 1$ for every $x\in […]

$\sigma$-algebras and independent stochastic processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. We consider a Wiener process $W$ with respect to his standard filtration $(\mathcal{F}_t^W)_{t \geq 0}$ and a process $X$ with his standard filtration $(\mathcal{F}_t^X)_{t \geq 0}$. We assume that $W$ and $X$ are independent. Let $(\mathcal{F}_t)_{t \geq 0}$ the filtration defined by $$\mathcal{F}_t = \sigma(\mathcal{F}_t^W \cup \mathcal{F}_t^X),$$ for […]

Lower Bound of Hausdorff Dimension of Cantor Set

Consider a Cantor set $E$ where the intervals at every level of the construction maintain a minimum spacing and have a finite number of intervals on each level. I have two questions regarding finding the lower bound of the Hausdorff dimension on such a set. Assume we have a collection of sets $\{U_i\}$ such that […]

Measurable function

How can I show that $f(x-y)g(y)$ is measurable on $\mathbb{R}^{2n}$ if $f,g$ are measurable on $\mathbb{R}^n$?

Can the unit interval be the disjoint union of countably many “super-dense” parts?

I’m curious about this question in the case where $f$ is not necessarily measurable. I think what it comes down to is this: Is there an $\varepsilon<1$ and a partition of $[0,1]$ in countably many parts such that every subset of $[0,1]$ with Lebesgue measure at least $\varepsilon$ intersects infinitely many of the parts? Clearly […]