I have the following result:
$$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$ where $a$ and $b$ are both non-negative.
Does it follow that $b(t) = g^{-1}(a(t))$? The problem is presence of the absolute value signs on $w(t)$, though the other functions are non-negative so this may be OK. Here $g$ is a real-valued function.