# An algebra that is not a $\sigma$-algebra

Let $${\cal C} \in P(X)$$ which $$X$$ is a non empty set and

$${\cal C}=\{A\in P(X); A \text{ or } A^c= X\smallsetminus A \}$$

Proof that $$\cal{C}$$ is a algebra but not a $$\sigma$$-algebra, and give an example.

I don't see what is the elements of set $${\cal C}$$ (Are $$A$$ only or $$A^c$$ only or both?), and I don't know how to solve it.

• could you explain better the definition of $\mathcal{C}$? Commented Sep 23, 2018 at 18:03
• it is in our textbook, me too I don't undertand it Commented Sep 23, 2018 at 18:05
• I think it's a typo that should instead be ${\cal C}=\{A\in P(X);\ A \text{ or } A^c \text{ is finite }\}$, where $A^c= X\smallsetminus A$ and $X$ is infinite. (They just omitted the "is finite".) Commented Sep 23, 2018 at 18:53

## 1 Answer

The class $$\mathscr{A}=\{F\subset \mathbb{N}: F \text{ is finite or } F^c \text{ is finite}\}$$is an algebra of subsets of $$\mathbb{N}$$ but not a $$\sigma$$-algebra. The let $$A_1, \ldots, A_n$$ sets in $$\mathscr{A}$$, then or every $$A$$ is finite or at last one $$A_i$$ is infinite (and therefore A^i^c is finite) so or the union is finite or the complement of the union has finite complement.

Clearly $$\mathscr{A}$$ is closed to complement.

The union of even numbers do not lies in $$\mathscr{A}$$, so $$\mathscr{A}$$ is not a $$\sigma$$-álgebra.

• When you take F is finite or F^cis finite is what meant by A or $A^C$ Commented Sep 23, 2018 at 18:01
• @MaryMaths I was not capable to understand the definition you give above of $\mathcal{C}$. So I give an example of an algebra which is not a $\sigma$-algebra. Commented Sep 23, 2018 at 18:02
• It is in our textbook,me too I don't undertand it, But in general what we search in general case. In others word how do you think to this example? Commented Sep 23, 2018 at 18:04
• I will explain in details in the answer. Could you provide me the name of your textbook? Commented Sep 23, 2018 at 18:05
• It is writen in french fsr.ac.ma/cours/maths/Ghanmi/Cours-SMA5%2017-18.pdf, Remarque 1.2.4. Commented Sep 23, 2018 at 18:07