What is the significance of $\mathscr F$ being algebra in defining measure? I know that if $S$ is a non-empty set and $\mathscr F$ is an algebra on $S$ then a function $\mu$ on $\mathscr F$ is called measure if
(i) $\mu(A)\in [0, \infty]$ for all $A \in\mathscr F$,
(ii) $\mu(\emptyset) = 0$,
(iii) $\mu\left(\displaystyle\bigcup_{n=1}^\infty A_n \right) = \displaystyle\sum_{n=1}^\infty \mu(A_n)$, where $\{A_n\}$ is a disjoint collection of members of $\mathscr F$, with $\left(\displaystyle\bigcup_{n=1}^\infty A_n \right) \in\mathscr F$.
My question is what is the significance of $\mathscr F$ being algebra? Can we take it to be semi-algebra?
(PS - I don't know how to make use of mathematical symbols here, please edit the question accordingly)
 A: Usually, one uses "measure" to talk about a measure over a $\sigma$-algebra. However, some people use only algebras, others are happy with semi-rings. And some people are happy calling "measure" what many would call an outer measure.
Why do people like semi-rings?
You look at $1 \times 1$ squares and you think: this has the area of a $1 \times 1$ square. Then, you look at rectangles and think... well I think I know the area of this $a \times b$ rectangle. It should be $ab$. Congratulations, you have defined area for a semi-ring or a semi-algebra, if you allow for infinite rectangles.
Semi-rings arise naturally, in my opinion, because of the rectangles (or cillinders).
Why do people like algebras?
Then, you realize that you can easily measure the area of finite unions of disjoint rectangles. Now, you have defined area for a ring... or an algebra, if you allow infinite rectangles.
People like algebras, in my opinion, because the definition is simpler. And since extending a $\sigma$-additive function over a semi-algebra to the generated algebra is very simple, some people tend (in my opinion) to favoring "algebras".
Why do people like $\sigma$-algebras?
Wouldn't it be nice if you could tell the area of a circle? Or, maybe, the area "below the graph" of a function...
But a cirlce is not in the algebra generated by the rectangles. You probably want to take limits of sets. You can approximate a circle using squares. Using this approximations in a consistent way, you can now measure the area of circles. The circle is in the $\sigma$-algebra.
At the end, what you really want (in my opinion), is a measure over a $\sigma$-algebra. There is a theorem (Carathéodory extension theorem) that states that you can always extend a "measure" defined over xxx to a "full measure" over the $\sigma$=algebra generated by xxx.
So, the questions sums up to whether you prefer to extend it form a semi-algebra xxx or from an algebra xxx.
