Intereting Posts

If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent
The number of bit strings with only two occurrences of 01
Inequality. $\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3$
Sum of digits and product of digits is equal (3 digit number)
Mathematical expression to form a vector from diagonal elements
Unusual mathematical terms
Sum of unit and nilpotent element in a noncommutative ring.
Derivative of a product and derivative of quotient of functions theorem: I don't understand its proof
Multi-pullbacks and the relative chinese remainder theorem
Closure of the span in a Banach space
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
How do I find partners for study?
Are sequences with Cesaro mean a closed subset of $\ell_\infty$?
Mid-point convexity does not imply convexity
For which $d \in \mathbb{Z}$ is $\mathbb{Z}$ a unique factorization domain?

Given a collection of sets $\mathcal{C}$ and $E$ an element in the $\sigma$-algebra generated by $\mathcal{C}$, how do I show that $\exists$ a countable subcollection $\mathcal{C_0} \subset \mathcal{C}$ such that $E$ is an element of the $\sigma$-algebra, $\mathcal{A}$ generated by $\mathcal{C_0}$?

The hint says to let $H$ be the union of all $\sigma$-algebras generated by countable subsets of $\mathcal{C}$….although I don’t know why.

- Differentiable function has measurable derivative?
- If $f$ and $g$ are integrable, then $h(x,y)=f(x)g(y)$ is integrable with respect to product measure.
- Continuity together with finite additivity implies countable additivity
- Measurability of a function defined on a product measure space, and related to a measurable function
- Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?
- Measurable rectangles inside a non-null set

- Smooth functions for which $f(x)$ is rational if and only if $x$ is rational
- Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$
- How to find all polynomials P(x) such that $P(x^2-2)=P(x)^2 -2$?
- Surface area with cavalier's principle
- The support of Gaussian measure in Hilbert Space $L^2(S^1)$ with covariance $(1-\Delta)^{-1}$
- $\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$
- What is the most rigorous proof of the irrationality of the square root of 3?
- Proof Check Lemma 2.2.10 in Tao
- Recurrence Relation for the nth Cantor Set
- Detail in Conditional expectation on more than one random variable

A good strategy would be to prove that $H$ is equal to $\sigma(\mathcal C)$, the $\sigma$-algebra generated by $\mathcal C$. It should be clear that $H \subset \sigma(\mathcal C)$. For the reverse inclusion, it is enough to show that $H$ is, in fact, a $\sigma$-algebra. Check each of the axioms! At some point, you will have to use the fact that a countable union of countable sets is countable.

- How to find the distance between two planes?
- $f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable
- Sequence of continuous linear functionals over a sequence of Hilbert spaces
- area of figure in sector of intersecting circles
- Proof by Cases involving divisibility
- If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
- A certain inverse limit
- Fractional Calculus: Motivation and Foundations.
- Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$?
- $TT^*=T^2$, show that $T$ is self-adjoint
- Introducing multiplication of cosets
- Integration of $e^{ax}\cos bx$ and $e^{ax}\sin bx$
- Exotic Manifolds from the inside
- If $\{a_1,\dots,a_{\phi(n)}\}$ is a reduced residue system, what is $a_1\cdots a_{\phi(n)}$ congruent to?
- Arbitrarily discarding/cancelling Radians units when plugging angular speed into linear speed formula?