Intereting Posts

If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$
Counting functions between two sets
Are the coordinate functions of a Hamel basis for an infinite dimensional Banach space discontinuous?
Null-recurrence of a random walk
Solving infinite sums with primes.
Character space of $L^{1} (\mathbb Z)$
$\int {e^{3x} – e^x \over e^{4x} + e^{2x} + 1} dx$
Finding Big-O with Fractions
What does a zero tensor product imply?
Sentence such that the universe of a structure has exactly two members
Fourier cosine transforms of Schwartz functions and the Fejer-Riesz theorem
Properties of Dedekind zeta function
$\mathbb{C}/(f,g)$ is an artinian ring, if $\gcd(f,g)=1$.
Prove that $(0)$ is a radical ideal in $\mathbb{Z}/n\mathbb{Z}$ iff $n$ is square free
How to prove that $\sum_{n=1}^{\infty} \frac{(\log (n))^2}{n^2}$ converges?

Please help me prove this:

Let $A_1,A_2,\ldots$ be subsets of $\Omega$. Prove that $A_n\to A$ if and only if $I_{A_n}(\omega)\to I_A(\omega)$ for every $\omega\in\Omega$ (so that convergence of sets is the same as pointwise convergence of their indicator functions).

Note: $I_A(\omega)=1$ if $\omega\in A$, and $0$ if $\omega\notin A$. Use in the proof that

$$\operatorname{lim\;inf}\limits_n\; x_n=\bigvee_{k=1}^\infty\bigwedge_{n=k}^\infty x_n\quad\text{

and }\quad\operatorname{lim\;sup}\limits_n\; x_n=\bigwedge_{k=1}^\infty\bigvee_{n=k}^\infty x_n.$$

Thank you very much!

- The method of proving the equality of integrals by showing they agree within $\epsilon$, for an arbitrary $\epsilon>0$
- Almost Everywhere Convergence versus Convergence in Measure
- Properties of $||f||_{\infty}$ - the infinity norm
- Why Are the Reals Uncountable?
- Uniform continuity and boundedness
- Difference between $\mathbb C$ and $\mathbb R^2$
- Prove that every isometry on $\mathbb{R}^2$ is bijective
- Properties of mollification for integrable functions
- Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$
- Show $\int_\pi^\infty \frac{dx}{x^2 \left( \sin^2(x) \right)^{1/3}}$ is finite using 1st semester measure theory

Let $\sigma=\langle A_n:n\in\Bbb N\rangle$ be a sequence of subsets of some set $\Omega$. A point $\omega\in\Omega$ is *eventually* in $\sigma$ if there is an $n_0\in\Bbb N$ such that $\omega\in A_n$ for all $n\ge n_0$, i.e., if $\omega$ is in each member of a ‘tail’ of the sequence. The point $\omega$ is *frequently* in $\sigma$ if for each $m\in\Bbb N$ there is an $n\ge m$ such that $\omega\in A_n$, i.e., if $\omega$ is in infinitely many members of the sequence. These terms provide an easy way to think and talk about the liminf and limsup of a sequence of sets: $\liminf_nA_n$ is the set of points of $\Omega$ that are eventually in $\sigma$, and $\limsup_nA_n$ is the set of points of $\Omega$ that are frequently in $\sigma$. This is quite easy to verify from the definitions. For example, $$\liminf_{n\in\Bbb N}A_n=\bigcup_{n\in\Bbb N}\bigcap_{k\ge n}A_k\;,\tag{1}$$ so $\omega\in\liminf_nA_n$ iff there is an $n\in\Bbb N$ such that $\omega\in\bigcap_{k\ge n}A_k$, which is the case iff $\omega\in A_k$ for each $k\ge n$: in short, $\omega\in\liminf_nA_n$ iff $\omega$ is eventually in $\sigma$. Similarly, $$\limsup_{n\in\Bbb n}A_n=\bigcap_{n\in\Bbb N}\bigcup_{k\ge n}A_k\;,\tag{2}$$ so $\omega\in\limsup_n A_n$ iff for each $n\in\Bbb N$ $\omega\in\bigcup_{k\ge n}A_k$, which is the case iff $\omega\in A_k$ for some $k\ge n$: $\omega\in\limsup_nA_n$ iff $\omega$ is frequently in $\sigma$.

It’s easy to check that $$\liminf_{n\in\Bbb N}I_{A_n}(\omega)=\bigvee_{n\in\Bbb N}\bigwedge_{k\ge n}I_{A_k}(\omega)\text{ for all }\omega\in\Omega$$ and $$\limsup_{n\in\Bbb N}I_{A_n}(\omega)=\bigwedge_{n\in\Bbb N}\bigvee_{k\ge n}I_{A_k}(\omega)\text{ for all }\omega\in\Omega$$ are simply restatements of $(1)$ and $(2)$ in terms of indicator functions. (E.g., $\omega$ is eventually in $\sigma$ iff $I_{A_n}(\omega)$ is eventually $1$.) Thus, the following statements are equivalent:

$$\begin{align*}&\lim_{n\in\Bbb N}A_n\text{ exists}\tag{3}\\&\liminf_{n\in\Bbb N}A_n=\limsup_{n\in\Bbb N}A_n\tag{4}\\&\liminf_{n\in\Bbb N}I_{A_n}(\omega)=\limsup_{n\in\Bbb N}I_{A_n}(\omega)\text{ for all }\omega\in\Omega\tag{5}\end{align*}$$

To finish the proof, you need only show that $(5)$ is equivalent to

$$\lim_{n\in\Bbb N}I_{A_n}(\omega)\text{ exists for each }\omega\in\Omega\tag{6}$$

and then show that the limit in $(6)$ is the indicator function of the limit in $(3)$.

It’s all just a matter of translating between two ways of saying the same thing: $\omega\in A$ iff $I_A(\omega)=1$, and $\omega\notin A$ iff $I_A(\omega)=0$.

- Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?
- Hexagonal circle packings in the plane
- Prove that $N \subset Z(G)$
- Showing there is no ring whose additive group is isomorphic to $\mathbb{Q}/\mathbb{Z}$
- Germs of $C^\infty$ functions near $0$ vs. germs of infinitely differentiable functions at $0$
- Integrating each side of an equation w.r.t. to a different variable?
- Why $y=e^x$ is not an algebraic curve?
- Is every locally compact Hausdorff space paracompact?
- If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $$ is divisible by 56 but not 49.
- Finding the vector perpendicular to the plane
- Example of non-Noetherian non-UFD Krull domain?
- Proof by induction that $B\cup (\bigcap_{i=1}^n A_i)=\bigcap_{i=1}^n (B\cup A_i)$
- Evaluating $ \int_{-\pi /2014}^{\pi /2014}\frac{1}{2014^{x}+1}\left( \frac{\sin ^{2014}x}{\sin ^{2014}x+\cos ^{2014}x}\right) dx $
- Quickest way to understand Kruskal's Tree Theorem
- Maximal ideals of polynomial rings in infinitely many variables