Intereting Posts

Derivative of determinant of a matrix
Lost on rational and Jordan forms
Combinatorics: How to find the number of sets of numbers in increasing order?
Prove that $\lfloor \sqrt{p} \rfloor + \lfloor \sqrt{2p} \rfloor +…+ \lfloor \sqrt{\frac{p-1}{4}p} \rfloor = \dfrac{p^2 – 1}{12}$
How to project $x_2$ onto $u_1$
Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.
Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$
Numerical solution to a system of second order differential equations
Open mathematical questions for which we really, really have no idea what the answer is
Induced map of homology groups of torus
A multiple integral question
The two extreme cases of the product measure
What is sample variance of sample variance, and what is theoretical sampling distribution?
Schröder-Bernstein for abelian groups with direct summands

Could you guide me how to prove that any monotone function from $R\rightarrow R$ is Borel measurable?

Should we separate the functions into continuous and non-continuous? How to prove for not continuous points?

Thanks for your help

- Calculate integral with cantor measure
- Relation between the two probability densities
- Confusions about Radon-Nikodym derivative and dominating measures
- Integration in respect to a complex measure
- Is there an example of a sigma algebra that is not a topology?
- Do these $\sigma$-algebras on second countable spaces coincide?

- Seeking a layman's guide to Measure Theory
- Volume form on Hamiltonian level surface
- Why do we essentially need complete measure space?
- Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$
- The Laplace transform of the first hitting time of Brownian motion
- Measurable function
- $f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable
- Integration by parts for general measure?
- How to find an irrational number in this case?
- An uncountable family of measurable sets with positive measure

Hint: If $f$ is monotone, then, for every real number $x$, the set $f^{-1}((-\infty,x])=\{t\mid f(t)\leqslant x\}$ is either $\varnothing$ or $(-\infty,+\infty)$ or $(-\infty,z)$ or $(-\infty,z]$ or $(z,+\infty)$ or $[z,+\infty)$ for some real number $z$.

To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-\infty,x])$, then show that, for every $v\leqslant u$, $v$ is also in $f^{-1}((-\infty,x])$.

- Primary decomposition of a monomial ideal
- Different types of domains $\Omega \subset \mathbb{R}^n$ in PDEs
- Show that the lexicographic order topology for $\mathbb{N}\times \mathbb{N}$ is not the discrete
- $\pi_n(X^n)$ free Abelian?
- Infinite Sum of Sines With Increasing Period
- Probability that all $n$ distinct points drawn independently on $$ are not “too close”
- Are these two definitions of $EG$ equivalent?
- Intuition of upper & lower bound of sequence of subsets.
- Show that the arc length of a curve is invariant under rigid transformation.
- Automorphism of an elementary extension of a structure that moves an undefinable element
- $\mathbb{F}_p/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
- Prove that $x^3-2$ and $x^3-3$ are irreducible over $\Bbb{Q}(i)$
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- What is $\gcd(0,a)$, where a is a positive integer?
- Proof that Fibonacci Sequence modulo m is periodic?