Could someone please help me find my way about this statement:

Let A be a sub-sigma-algebra of the Borel sets of the torus; A is
invariant to the circle action (x,y) $\rightarrow$ (x,y+g) mod 1. Then
any square intergrable, A-measurable function f has the following
representation $f=\sum\limits_{k \in Z}f^{k}$, where $f^{k}(x,
y+g)=e^{2 \pi ikg}f^{k}(x,y)$ for all $g\in[0,1)$.

I do not understand:

- What does it mean that a sigma-algebra is action invariant?
- Why do the functions in the basis (f^{k}) let g come out as an
exponential multiplier.

Would you refer me to the main theorems that are implied in that
statement and if possible a reference as to where I could find them?