Intereting Posts

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?
How one can obtain roots at the negative even integers of the Zeta function?
How do we find specific values of sin and cos given the series definition
A logic that can distinguish between two structures
Detail of the proof that the cardinality of a $\sigma$-algebra containing an infinite number of sets is uncountable
Construct a monotone function which has countably many discontinuities
An alternative proof of Cauchy's Mean Value Theorem
What does it mean for a matrix to be orthogonally diagonalizable?
What was the motivation for the complex plane?
Question about sum of l'p spaces
$\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}=\prod_{k=1}^{n}\bigl(1-\frac{1}{2k}\bigr)$
Isosceles triangle
Congruence Class $_5$ (Equivalence class of n wrt congruence mod 5) when n = $-3$, 2, 3, 6
$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real
A Hamel basis for $l^{\,p}$?

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.

- Omitting the hypotheses of finiteness of the measure in Egorov theorem
- Absolutely continuous maps measurable sets to measurable sets
- Given $f\notin L^p$ find $g\in L^q$ s.t. $fg\notin L^1$
- Lebesgue measure as $\sup$ of measures of contained compact sets
- Measure Theory - Absolute Continuity
- How to show that the Dini Derivatives of a measurable function is measurable?
- Positive outer measure set and nonmeasurable subset
- Measurability of almost everywhere continuous functions
- Integral of Schwartz function over probability measure
- An application of the General Lebesgue Dominated convergence theorem

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.

- Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$
- Show that a finite group with certain automorphism is abelian
- Countable infinite direct product of $\mathbb{Z}$ modulo countable direct sum
- Is the $n$-th prime smaller than $n(\log n + \log\log n-1+\frac{\log\log n}{\log n})$?
- How to calculate a heading on the earths surface?
- Is a measure for a sigma algebra determined by its values for a generator of the sigma algebra?
- How to prove that a connected graph with $|V| -1= |E|$ is a tree?
- How many groups of order $815,409=3^2\times 7^2\times 43^2$ are there?
- Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?
- Roadmap to study Atiyah–Singer index theorem
- Dropping the “positive” and “decreasing” conditions in the integral test
- identity of $(I-z^nT^n)^{-1} =\frac{1}{n}$
- Motivation for Eisenstein Criterion
- Inequality of length of side of triangle
- Generating Functions- Closed form of a sequence