Intereting Posts

A contest question on probability
A normal matrix with real eigenvalues is Hermitian
Find the integer $x$ such $x^6+x^5+x^4+x^3+x^2+x+1=y^3$
Product of Principal Ideals when $R$ is commutative, but not necessarily unital
QM-AM-GM-HM proof help
Finding the limit of roots products $(\sqrt{2}-\sqrt{2})(\sqrt{2}-\sqrt{2})(\sqrt{2}-\sqrt{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt{2})$
Expectation of ratio of sums of i.i.d. random variables. What's wrong with the simple answer?
Factor group of a center of a abelian group is cyclic.
Does the power spectral density vanish when the frequency is zero for a zero-mean process?
How to differentiate CDF of Gamma Distribution to get back PDF?
Free Throw Probability and Expected Number of Points
Sturm-Liouville Questions
Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?
Quadratic residues and representations of integers by a binary quadratic form
2013 Putnam A1 Proof understanding (geometry)

The Radon–Nikodym theorem states that,

given a measurable space $(X,\Sigma)$, if a $\sigma$-finite measure $\nu$ on $(X,\Sigma)$ is absolutely continuous with respect to a $\sigma$-finite measure $\mu$ on $(X,\Sigma)$, then there is a measurable function $f$ on $X$ and taking values in $[0,\infty)$, such that

$$\nu(A) = \int_A f \, d\mu$$

- Finitely but not countably additive set function
- Do there exist two singular measures whose convolution is absolutely continuous?
- Construction of a Borel set with positive but not full measure in each interval
- Assume that $ f ∈ L()$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.
- Examples of measurable and non measurable functions
- Convolution of an integrable function of compact support with a bump function.
for any measurable set $A$.

$f$ is called the Radon–Nikodym derivative of $\nu$ wrt $\mu$.

I was wondering in what cases the concept of Radon–Nikodym derivative and the concept of derivative in real analysis can coincide and how?

Thanks and regards!

- Show that $x+e^{-Bx^2}\mbox{cos}(x)$ has only one root over all reals ($B>0$).
- Convergence in measure implies convergence almost everywhere of a subsequence
- Let the function $f: \to \mathbb R$ be Lipschitz. Show that $f$ maps a set of measure zero onto a set of measure zero
- Dirichlet's function expressed as $\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)$
- Existence of the limit of a sequence
- Cauchy-Formula for Repeated Lebesgue-Integration
- Cantor diagonalization method for subsequences
- Approximating a $\sigma$-algebra by a generating algebra
- Prove: bounded derivative if and only if uniform continuity
- totally bounded, complete $\implies$ compact

Consider absolute continuity of functions as defined in Royden’s *Real Analysis*. These functions are integrals of their derivatives and they are, in fact Radon-Nikodym derivatives.

- Orthogonal Latin Square 6*6
- Analyzing whether there is always a prime between $n^2$ and $n^2+n$
- Thinkenning a Renewal Process
- A Vitali set is non-measurable, direct proof, without using countable additivity
- What is the use of the Dot Product of two vectors?
- Finding range of a linear transformation
- Can the inequality $\sum\limits_{i=1}^n\frac{1}{\sqrt{i}} < 2\sqrt{n} – 1$ be proved without induction?
- Kronecker product and outer product confusion
- Prob. 9, Chap. 6, in Baby Rudin: Integration by parts for improper integrals
- Question about homomorphism of cyclic group
- Show that every graph $G$ has a bipartite subgraph with at least half of the edges of $G$
- Explaining $\cos^\infty$
- Closed form of $\int_0^{\pi/2} \frac{\arctan^2 (\sin^2 \theta)}{\sin^2 \theta}\,d\theta$
- Proving that every vector space has a norm.
- Generating Functions- Closed form of a sequence