Why do we call it a $\sigma$-algebra?

In simple terms, a $\sigma$-algebra is the collection of all of the things we know how to measure. Why don't we call it something that more directly suggests this, for example a 'measure space?'

Solution 1:

Sure, the $\sigma$-algebra $\mathcal A$ in a measure space $(X,\mathcal A,m)$ is exactly the collection of subsets $B$ of $X$ such that $m(B)$ exists but to define $\sigma$-algebras as "the collection of all of the things we know how to measure" is to put the cart before the horse since $\sigma$-algebras exist without measure. In fact a $\sigma$-algebra $\mathcal A$ on a set $X$ is a subset of $2^X$ with the properties you might know--then one says that $(X,\mathcal A)$ is a measurable space.

Thus, "measure spaces" are triplets $(X,\mathcal A,m)$, whose second bit $\mathcal A$ is a $\sigma$-algebra. Note that, in general, tons of measures $m$ can be defined on a same measurable space $(X,\mathcal A)$ hence to call the part $\mathcal A$ by the name of the whole $(X,\mathcal A,m)$ would be absurd.

About terminology now: Greek letters $\sigma$ and $\delta$ (small sigma and small delta) are often used when countable unions and countable intersections are involved. Topologists call $F_\sigma$ every countable union of closed sets in a topological space (F standing possibly for the French word fermé, closed) and $G_\delta$ every countable intersection of open sets (G standing for the German word Gebiet, domain, connected open set). The letters $\sigma$ and $\delta$ are often given as Greek abbreviations of German words: $\sigma$ as S in Summe for sum (in the sense of sum of sets, that is, union) and $\delta$ as D in Durchschnitt for intersection, both countable. Thus, in the context of measure theory, the letter $\sigma$ refers to the stability of a collection of subsets by countable union.