Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$$ S_n = \mathscr P (\{ -n, -n+1, \ldots, n-1, n\}) $$ $$ R_n = \{r : \Omega - r \in S_n\} $$ $$ T_n = S_n \cup R_n$$

I need to check whether

  1. $T_n$ is an algebra, semi-algebra or sigma algebra.

  2. $T_n \subset T_{n+1}$.

  3. If $T = \bigcup_n T_n$, whether $T$ is algebra, semi-algebra, sigma algebra.

I considered an example for this:

Let $ S_1 = \mathscr P \{-1, 0, 1\} $ $$ S_1 = \{\{-1\}, \{0\}, \{1\} ,\{-1,0\}, \{0,1\}, \{-1,1\}, \{-1,0,1\},\{ \varnothing\}\}$$

So, for 1) I feel that $T_n = \Omega$, so it can either be any of the algebras.

Please advise.

share|cite|improve this question
Which part of the definitions are you having trouble checking? – Trevor Wilson Sep 16 '12 at 15:43
Still, I want to check my idea for 1) is correct. – mathguy Sep 16 '12 at 16:05
What is $\Omega$? Is it $\mathbb{Z}$? And do you mean that $T_n = \mathscr{P}(\Omega)$ for all $n$? If so, why? – Trevor Wilson Sep 16 '12 at 16:47
Does what $\Omega$ is matter here? Since $T_n = S_n \cup R_n$ doesn't $T_n$ automatically becomes $\Omega$. – mathguy Sep 16 '12 at 16:58
I think you may be confused about what the definitions in the problem mean. The symbol $\mathscr{P}$ means "power set," so for example $S_1$ consists of all eight subsets of the set $\{-1,0,1\}$. – Trevor Wilson Sep 16 '12 at 17:08
  1. $T_n$ is stable by complementation (almost by definition) and by intersections: if $A, B\in T_n$ and $A,B\in S_n$ then $A\cap B\in S_n$, if $A,B\in R_n$ then $A\cap B\in R_n$ and if $A\in R_n$ and $B\in S_n$ then $A\cap B=\emptyset$. So $T_n$ is a semi-algebra and since $\Omega\in T_n$, an algebra. $T_n$ is a $\sigma$-algebra as a finite algebra.

  2. Since $S_n\subset S_{n+1}$, if $A\in T_n$, either $A\in S_n$, then $A\in S_{n+1}\subset T_{n+1}$ or $A\in R_n$, then $\Omega\setminus A\in S_n\subset S_{n+1}$ hence $A\in T_{n+1}$, which proves inclusion.

  3. An increasing sequence of (semi-)algebras is a (semi-)algebra so $T$ is a (semi-)algebra. It's not a $\sigma$-algebra as $\{2k,k\in\Bbb Z\}$ is a countable union of elements of $T$ but not an element of $T$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.