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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.

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  • $\begingroup$ could you explain better the definition of $\mathcal{C}$? $\endgroup$
    – Eduardo
    Commented Sep 23, 2018 at 18:03
  • $\begingroup$ it is in our textbook, me too I don't undertand it $\endgroup$
    – Mary Maths
    Commented Sep 23, 2018 at 18:05
  • $\begingroup$ 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".) $\endgroup$
    – r.e.s.
    Commented Sep 23, 2018 at 18:53

1 Answer 1

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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.

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  • $\begingroup$ When you take F is finite or F^cis finite is what meant by A or $A^C$ $\endgroup$
    – Mary Maths
    Commented Sep 23, 2018 at 18:01
  • $\begingroup$ @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. $\endgroup$
    – Eduardo
    Commented Sep 23, 2018 at 18:02
  • $\begingroup$ 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? $\endgroup$
    – Mary Maths
    Commented Sep 23, 2018 at 18:04
  • $\begingroup$ I will explain in details in the answer. Could you provide me the name of your textbook? $\endgroup$
    – Eduardo
    Commented Sep 23, 2018 at 18:05
  • $\begingroup$ It is writen in french fsr.ac.ma/cours/maths/Ghanmi/Cours-SMA5%2017-18.pdf, Remarque 1.2.4. $\endgroup$
    – Mary Maths
    Commented Sep 23, 2018 at 18:07

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