Intereting Posts

If $H$ is a subgroup of $G$ of finite index $n$, then under what condition $g^n\in H$ for all $g\in G$
Proof: If $n=ab$ then $2^a-1 \mid 2^n-1$
Computing the dimension of a vector space of matrices that commute with a given matrix B,
Suprema proof: prove $\sup(f+g) \le \sup f + \sup g$
How to define Homology Functor in an arbitrary Abelian Category?
Is this equivalent to Cauchy-Schwarz Inequality?
Show that the Fubini Tonelli theorem does not work for this function
Short form of few series
Is the maximum function of a continuous function continuous?
Expected size of subset forming convex polygon.
What Topology does a Straight Line in the Plane Inherit as a Subspace of $\mathbb{R_l} \times \mathbb{R}$ and of $\mathbb{R_l} \times \mathbb{R_l}$
Structure of Finite Commutative Rings
$\operatorname{Spec}(k)$ has infinite points.
Complicated limit calculation
Poincaré series and short exact sequences

Let $Y$ be a collection of subsets of the set X. Show that for each $A \in \sigma(Y)$ there is a countable subfamily $B_0 \subset Y$ such that $A\in \sigma(B_0)$

My try: I look at $\cup B_i$ where $B_i$ is a countable subfamily of $Y$. And I want to show that $Y\subset \cup B_i \subset\sigma(Y)$. Both the $\cup B_i \subset\sigma(Y)$ and $Y\subset \cup B_i$ feels intuitive, but how do I write it out rigorously?

- Hausdorff Measure and Hausdorff Dimension
- Nonatomic vs. Continuous Measures
- Simple inequality for measures
- Alternative definition of $\|f\|_{\infty}$ as the smallest of all numbers of the form $\sup\{|g(x)| : x \in X \}$, where $f = g$ almost everywhere
- continuity of norms with respect to $p$
- Exercise on convergence in measure (Folland, Real Analysis)

- Proof that the Lebesgue measure is complete
- Characterization of Almost-Everywhere convergence
- Can anyone clarify why this is?
- $L^p$ implies polynomial decay?
- Prove that a set $A$ is $\mu^\star$ measurable is and only if $\mu^\star (A) = l(X) - \mu^\star(A^{c})$
- Is the intersection of an arbitrary collection of semirings a semiring?
- Basic Geometric intuition, context is undergraduate mathematics
- Measure of Image of Linear Map
- Understanding the measurability of conditional expectations
- The set of convergence of a sequence of measurable functions is measurable

The way to prove it is to consider the set $\mathcal{C}$ of all $X\in\sigma(Y)$ such that for some countable $B_0\subseteq Y$, $X\in\sigma(B_0)$. Clearly, $\mathcal{C}$ is closed under complements and contains every element of $Y$. If you can show that $\mathcal{C}$ is closed under countable unions, which follows from the fact that countable unions of countable sets are countable, you have established that $\mathcal{C}$ is a $\sigma$-algebra satisfying $$Y\subseteq\mathcal{C}\subseteq\sigma(Y),$$

and hence $\mathcal{C}=\sigma(Y)$.

- Calculating $\prod (\omega^j – \omega^k)$ where $\omega^n=1$.
- Proof that if $f$ is integrable then also $f^2$ is integrable
- Equivalence of weak forms of Hilbert's Nullstellensatz
- Trying to get the point into a line at a given distance
- Simple explanation for Hypergeometric distribution probability
- Polynomials, finite fields and cardinality/dimension considerations
- How to solve this sequence $165,195,255,285,345,x$
- Question about Holomorphic functions
- What is the Kolmogorov Extension Theorem good for?
- If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$
- Maximum likelihood estimator of $\theta>-1$ from sample uniform on $(0,\theta+1)$
- transform integral to differential equations
- Free product as automorphism group of graph
- Why does the Fibonacci Series start with 0, 1?
- Representing negative numbers with an infinite number?