I don't understand the definition of an algebra and $\sigma$-algebra Here is a definition I have for an algebra:
A family $f$ of subsets of the set $\Omega$ is called an algebra if it satisfies:


*

*$\emptyset, \Omega \in f$

*If $A \in f$ then $A^c \in f$

*If $A,B \in f$ then $A \cup B \in f$


What I don't understand about this definition is that if $f$ is a family of subsets of a set $\Omega$ then how can $\Omega$ be in $f$? I just can't follow that. I also do not understand the motivation of this definition.
I have taken a look at an example online and I have found this:
Let $ \Omega = [0,1]$ and $f$ the family of all subsets of $[0,1]$ w hich can be expressed as a finite union of intervals. Then $f$ is an algebra, but not a $\sigma$-algebra. 
The page did not expand at all and simple just stated it was an algebra, and not a $\sigma$-algebra. I feel as though I'm missing something obvious, I cannot understand how $\Omega \in f$.

Another way to think about it is $f$ is a subset of power set of $\Omega$.
For any set $X$, the power set of $X$, denote $P(X)$ or $2^X$, is the collection of all possible subsets of $X$.
For example,
$X=\{a\}, P(X)=\{\emptyset, \{a\}\}$
$X=\{a, b\}, P(X)=\{\emptyset, \{a\}, \{b\}, \{a, b\}\}$
$X=R$, $P(X)=$ all subsets of real numbers
etc.
