Linked Questions

2
votes
2answers
114 views

Trying to find two functions $\varphi$ measurable and $f$ continuous such that $\varphi\circ f$ isn't measurable [duplicate]

A set $E \subseteq \mathbb{R}$ is measurable if for any subset $A\subset\mathbb{R}$, $$\mu(A)=\mu(A\cap E)+\mu(A\cap E^c).$$ where $\mu$ is the Lebesgue outer measure. A function $\varphi : \mathbb{...
0
votes
2answers
122 views

Prove a composition of two functions is meaurable [duplicate]

Let $f,g:[0,1]\rightarrow [0,1]$ be measurable functions.Is $g\circ f$ measurable or not? The composition is definitely measurable from the axiom definition of measurable function. But if we want to ...
6
votes
2answers
3k views

composition of Lebesgue measurable functions is not Lebesgue mesurable

Let $f,g:\mathbb{R}\longrightarrow \mathbb{R}$ be Lebesgue measurable. If $f$ is Borel measurable, then $f\circ g$ is Lebesgue mesuarable. In general, $f\circ g$ is not necessarily Lebesgue measurable....
6
votes
1answer
1k views

$f$ is measurable and $g$ is monotonic continuous, is $f \circ g$ Lebesgue measurable?

Let $f$ be a measurable function on real numbers and $g$ is a monotonic continuous function on real numbers. Is the function composition $f \circ g$ Lebesgue measurable? Thanks.
4
votes
2answers
128 views

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function, is $f \circ \sin$ measurable?

If $f \circ \sin$ is measurable, then we need to show that $(f \circ \sin)^{-1}(B) = \sin^{-1}(f^{-1}(B))$ is measurable for every Borel set $B$, it is sufficient to show that $\sin^{-1}$ takes a ...
0
votes
2answers
204 views

How do I show this function is measurable?

Let $h:[0,\infty)\to\mathbb{R}$ be a monotone function, with $\int_0^\infty |h(x)|x^2\,dx<\infty$. And let $f:\mathbb{R}^3\to\mathbb{R}$ with $f(x)=h(|x|)$ for all $x$. Prove that $f$ is (...
0
votes
1answer
321 views

measurability of magnitude and sign function of complex function

I am trying to prove the following: (I) Let $f:X\rightarrow \mathbb{C}$ (i.e. complex space) and $f=u+iv$ where $u$ is the real part and $v$ is the imaginary part. If $f$ is measurable, then the ...
2
votes
2answers
81 views

If $B$ Borel measurable, $x \in \mathbb{R}$, then $B + \{x\}$ Borel measurable

I imagine that this is pretty obvious, but I'm missing something. It's part of a larger proof to show that if $B$ Borel, $A$ countable, then $B+A$ Borel. If I can get $B + \{x\}$ Borel, the rest ...
1
vote
2answers
131 views

Given two stochastic processes on a probability space, will their compound process be a valid stochastic process on the same probability space?

Let the stochastic process $M=(M_t, t\ge 0)$ and the stochastic pathwise continuous increasing process $Y=(Y_t,t\ge 0)$ be defined on the probability space $(\Omega, \mathcal F, P)$. Will the compound ...
2
votes
2answers
45 views

Why can I not use this alternative, simpler way of showing that $\frac{1}{|a|}f(\frac{x-b}{a})$ is Borel measurable

Let $X$ be a real random variable and $f$ is its density w.r.t. the Lebesgue measure. As a background, I was asked to show the density of $Y:=aX+b$ exists and is $g(x):=\frac{1}{|a|}f(\frac{x-b}{a})$....
1
vote
2answers
57 views

Composition of continuous and Lp function is essentially bounded?

Let $f:\Omega \rightarrow \mathbb{R} $ be a function in $L^p(\Omega)$ for some $1\leq p < \infty$ and $g: \mathbb{R} \rightarrow[a,b]$ continuous, with $a,b\in \mathbb{R}$. Do we have $g \circ f \...
0
votes
0answers
42 views

Example of function $f,g$ s.t. $f$ continuous, $g$ measurable and $f\circ g$ not measurable.

In my course it's written that if $f$ measurable and $g$ continuous then $g\circ f$ is measurable. But is there example of functions $f$ and $g$ s.t. $f$ continuous, $g$ measurable and $g\circ f$ not ...