I am reading the appendix of Ballman's "Lectures on Spaces of Nonpositive Curvature", which proves the ergodicity of geodesic flows of compact manifolds with strictly negative sectional curvature.
In Proposition 2.6, for $\phi$ a continuous flow on a compact metric space $X$ preserving a finite measure $\mu$ which is positive on open sets, the proof begins by assuming without loss of generality that $f$, a $\phi$-invariant measurable function, is bounded below by 0.
Why is this reasonable?
The goal is to prove that $f$ is constant almost everywhere (a.e.).
Let's break the function $f$ into a difference of two non-negative, measurable, $\phi$-invariant functions $f = f_+ - f_-$, where $$f_+(x) = \max\{0,f(x)\}, \quad f_-(x) = \max\{0,-f(x)\} $$ Each of the two functions $f_-,f_+$ is bounded below by $0$.
So if we can prove the theorem for those functions which are bounded below by $0$, it will follow that each of $f_-,f_+$ is constant (a.e.), say $f_+(x)=a$ (a.e.) and $f_-(x)=b$ (a.e.).
From this it will follow that $f(x) = a-b$ (a.e) is constant almost everywhere.