Intereting Posts

Decomposition as a product of factors
Exact sequence and torsion
When does it help to write a function as $f(x) = \sup_\alpha \phi_{\alpha}(x)$ (an upper envelope)?
$x^p -x-c$ is irreducible over a field of characteristic $p$ if it has no root in the field
Counting strings containing specified appearances of words
If $n$ is an odd pseudoprime , then $2^n-1$ is also odd pseudoprime
Am I allowed to use distributive law for infinitely many sets?
The expected area of a triangle formed by three points randomly chosen from the unit square
Proof that monotone functions are integrable with the classical definition of the Riemann Integral
Showing $(P\to Q)\land (Q\to R)\equiv (P\to R)\land$ {without truth table}
Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$
Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution
Axiom of Choice and finite sets
Proof of upper-tail inequality for standard normal distribution
Distinct digits in a combination of 6 digits

Given an example of a continuous function $\phi$ and a lebesgue measurable function $F$. both defined on $[0,1]$, such that $F\circ\phi$ is not lebesgue measurable.

- Can a continuous surjection between compacts behave bad wrt Borel probability?
- Hahn-Banach to extend to the Lebesgue Measure
- Integration with respect to Dirac measure
- From distribution to Measure
- How do I go about proving da db/a^(-2) is a left Haar measure on the affine group?
- Calculate an integral in a measurable space
- Meaning of non-existence of expectation?
- Confused about Cantor function and measure of Cantor set
- How to compute Riemann-Stieltjes / Lebesgue(-Stieltjes) integral?
- Compact inclusion in $L^p$

Here is an example, taken from here:

Let $f:[0,1]\to \mathbb R$ be the Cantor function. It has range $[0,1]$.

Define the function $g:[0,1]\to \mathbb R$ by $g(x)=x+f(x)$.

The function $g$ has range $[0,2]$, is continuous and injective on $[0,1]$

and has a continuous inverse on its range. It also maps the Cantor set

$C\subset [0,1]$ to a set of masure $1$, i.e, $m(g(C))=1$.

Since $m(g(C))=1$, there exists a nonmeasurable set $D$ contained in $g(C)$.

Then $E= g^{-1}(D)$ is contained in $C$ so it has measure zero.

Define $h$ to be the characteristic function of $E$. Then $h$ is measurable on

$[0,1]$ but $h(g^{-1})$ is the nonmeasurable characteristic function of the

non-measurable set $D$.

As pointed out, originally this example was taken from the Dover book “Counterexamples in Analysis” by Gelbaum/Olmsted. Hope this helps.

- Explanation of Maclaurin Series of $x^\pi$
- Prime gaps with respect to the squared primes
- Unique perpendicular line
- Finding the last two digits of a number by binomial theorem
- An irreducible $f\in \mathbb{Z}$, whose image in every $(\mathbb{Z}/p\mathbb{Z})$ has a root?
- Travelling salesman – Linear Programming
- Lower bound for finding second largest element
- Bounded, non-constant harmonic functions: how far are they from existing?
- Sylow questions on $GL_2(\mathbb F_3)$.
- An idempotent operator is compact if and only if it is of finite rank
- Is norm non-decreasing in each variable?
- Is there a domain which is not UFD but has a maximal principal ideal?
- Primal and dual solution to linear programming
- The number of ways to order 26 alphabet letters, no two vowels occurring consecutively
- Is there a way to see this geometrically?