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Given a sequence of reals $(a_n)_{n > 0}$, let $f: [0, 1] \to R$ be the generalised raindrop function defined: $f(x) = a_q$ if $x$ is rational, with denominator $q$ in lowest form; $0$ otherwise. Qu …

Here all functions are from $[0, 1] \to \mathbb R$. Let $f_i$ be a sequence of continuous functions such that there exists some $M > 0$ such that $|f_i| < M$ for all $i$. Does there always exist some …

Does there exist a measurable subset $T$ of $[0, \infty)$ with finite measure and some $\epsilon > 0$ such that for every $r$ with $0 < r < \epsilon$, $nr$ is in $T$ for infinitely many positive integ …

Given a locally integrable function $f: \mathbb R \to \mathbb R$, define $ K: \mathbb R \times \mathbb R+ \to \mathbb R$ by $$ K(x, r) := \begin{cases} 1, & \text{if }f(x) > \dfrac{1}{2r}\displayst …

I have two functions, and I want to combine them to define a certain function. Suppose for every fixed $e$ in $(0, ∞)$, we have a function $g_e (x): \mathbb{R} \to [0,\infty]$ that is well defined a. …

Suppose $f(x, e): \mathbb{R} \times (0, \infty)\to [0,\infty]$ is right continuous in $x$, and monotone increasing in $e$. Is $f$ jointly measurable?

Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and …

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$. Let $f$ be measurable. For every $e$ in …

Given a a.e. finite measurable function $ \mathbb R^n \to \mathbb R$, define the essential modulus of continuity, $M(f): \ \mathbb R^n \times \mathbb R+ \to \mathbb R$ by $$ M(f) (x, e)=\sup_{m(A) = 0 …

Let $X$ be a measure space. Let $S_j$, $j \in \mathbb N$ be an increasing sequence of $\sigma$-algebras on $X$ such that $S := \bigcup_{j \geq 0} S_j$ is a $\sigma$-algebra. For every $j$, let $\mu_j$ …

Call a function $f: \mathbb R+ \to \mathbb R$ essentially $C^\infty$ if there exists a sequence $f_n$ $(n \geq 0)$ such that each $f_n$ is differentiable a.e., $f_0 = f$ a.e., and $f_n’$ is equal to $ …

Note: Functions that differ on a null set are not identified. Consider the space of measurable functions $[0, 1] \to [0, 1]$ that are continuous exactly on a set of Lebesgue measure $r$ , $0 < r < 1$ …

A subset $E$ of $\mathbb R$ is said to contain arbitrarily long arithmetic progressions, if for every natural $n$, there exists $a, d \in R, d$ nonzero, such that $a + kd$ is in $E$ for all natural $k …

Let $X$ be a measure space, and suppose $\mu_i$ are probability measures on $X$ that are absolutely continuous with respect to another probability measure $\mu$. Is strong convergence of $\mu_i$ to $\ …

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$. Let $f$ be measurable. For every $e$ in …