I am reading the elements of integration by Robert Bartle.
In chapter 2 the definition of a $\sigma$ algebra is given like this.
A family $\sum$ of subsets of a set X is said to be a $\sigma$ algebra if
(i) $\phi$, X belongs to $\sum$.
(ii)If A belongs to $\sum$ then the compliment $X \backslash A$ belongs to $\sum$
(iii) If $(A_n)$ is a sequence of sets in $\sum$ then union $\cup^\infty_{n=1}A_n$ belongs to $\sum$
Then in chapter 9, the definition of an algebra is given like this.
A family $\sum$ of subsets of a set X is said to be an algebra if
(i) $\phi$, X belongs to $\sum$.
(ii) If E belongs to $\sum$ then its complement $X \backslash E$ also belongs to $\sum$.
(iii) If $E_1,...,E_n)$ belongs to $\sum$ then their union $\cup^n_{j=1} E_j$ also belongs to $\sum$.
What is the difference between these two definitions. What is the difference between a $\sigma$ algebra and an algebra?
