A difficult question about almost everywhere valid properties Let $\mu$ be a measure and $[f]\in L^2(\mu)$, i.e. $$[f]=\left\{g\in\mathcal{L}^2(\mu):f\equiv g\;\;\;\mu\text{-almost everywhere}\right\}$$ Moreover, let $x^+:=\max(x,0)$ for $x\in\mathbb{R}$. Suppose we want to prove the existence of a function $h$ with a certain property and can show, that there is a $g\in [f]$, which depends on a parameter $p$, such that $g^+$ satisfies the property (which is independent of $p$). So, we would set $$h:=g^+$$ to complete the proof. Can we say $$h\ge 0$$ or only $$h\ge 0\;\;\;\mu\text{-almost everywhere}\;?$$

You need to give more details about the property you want $h$ to satisfy and the role played by the parameter.
If the property only depends on $[h]$, then you can always change $h$ to be zero where it used to be negative, provided that $h\geq0$ almost everywhere.
In your example, $g^+$ is positive everywhere (it is defined pointwise), so $h=g^+\geq0$ everywhere.
Having a parameter seems to make no difference, assuming I understand the question correctly.
