I find it difficult to understand why the ‘size’ of the set of rational numbers in an interval such as [0,1] is zero. I know that there are way more irrational numbers than rational numbers such that m(set of irrational numbers) = 1 and as such m(set of rational numbers)=0. But I still find it […]

Consider the measurable space $(\mathbb{R}, \Sigma)$, where $$\Sigma := \{ A \subset \mathbb{R} \,:\, A \text{ is countable or } A^c \text{ is countable}\}.$$ Proving this is indeed a $\sigma$-algebra is easy: The countable union of countable sets is again countable; if the countable union contains at least one cocountable set, the coset of this […]

Suppose $\sum^{\infty}a_{i}1_{A_{i}}\geq \sum^{\infty}b_{i}1_{B_{i}}$, where $a_{i},b_{i}\geq 0$ and the sets possibly intersect i.e. $A_{i}\cap A_{j}\neq \varnothing $ and same with $B_{i}$. Is it true that we can write $\sum^{\infty}a_{i}1_{A_{i}}-\sum^{\infty}b_{i}1_{B_{i}}=\sum^{\infty}c_{i}1_{C_{i}}$ for $c_{i}\geq 0$? If we have disjointness i.e. $A_{i}\cap A_{j}=\varnothing $ and same for $B_{i}$ then yes. My concern is that when I proved it the general […]

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space. Take $f,g \in L^1(\mu)$. Prove that $\sqrt{f^2+g^2}$ and $\sqrt{\vert fg\vert}$ are in $L^1(\mu)$. First, I prove that $h = \max\{f,g\} \in L^1(\mu)$. Let $A = \{x \in \mathcal{X} \mid f(x) \geq g(x)\}$. Then: $$ \begin{align} \int_{\mathcal{X}} h \ \mathrm{d}\mu &= \int_A h \ \mathrm{d}\mu + \int_{A^c} h […]

It seems there should exist a non-measurable bijection $f: \mathbb{R}\rightarrow \mathbb{R}$. And thus we can obtain a non-measurable graph on $\mathbb{R}^2$ which intersects every horizontal or vertical line at precisely one point. Does anyone know how to “construct” such a map? Can it be further made into a automorphsim (w.r.t the addtive group or field […]

Let $f:X\rightarrow \overline{\mathbb{R}}$ be $\mathcal{A}$-measurable and let $B\in \mathcal{A}$. I would like to show that $\chi_{B}f $ is $\mathcal{A}$-measurable. I want to find the set $[\chi_{B}f > a ]$ for all $a \in \mathbb{R}$ , how to do that?

Let us consider the following integral: $$I(x)=\int_\Omega f(x, \omega)\, d\omega, $$ where $\Omega$ is a measure space and $f\colon \mathbb{R}\times \Omega \to \mathbb{R}$ is such that $f(x, \cdot)\in L^1(\Omega)$ for all $x$. When can we differentiate $I$? A dominated convergence argument gives the following result. Proposition. If For almost all $\omega\in \mathbb{\Omega}$, $$f(\cdot, \omega)\ \text{is […]

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 […]

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 […]

I was thinking if it is possible to come up with a $\sigma$-finite measure on $\mathbb R$ which is positive on any uncountable set. I think that I have a proof that there is no such measure – but I am not sure if it is formal enough. The proof: let $\mu$ be such measure, […]

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