Calling $\int_{a}^{b*} f = \inf\limits_{P \in P([a,b])} U_{f,P}$
Where $U_{f,P} = \sum\limits_{k=1}^{n} \sup\limits_{x \in [x_{k-1},x_{k}]}f(x) \cdot ( x_{k}-x_{k-1})$ is the upper sum on a given partition, and $P([a,b])$ the set of all the possible partions of $[a,b]$.
In this context we know that
$$\int_{a}^{b*} (f+g) \leq \int_{a}^{b*} f + \int_{a}^{b*} g$$
I was wondering wether i could find $f,g$ such that the inequality is strict,
Of course $f+g$ has not to be Riemann Integrable,otherwise we have equality by linerity,
But i was unable to find two explicit functions, i though taking $f=-g$, or $f = c \in \mathbb{R}$ trying to find $g$ would work, i even tried with Dirichlet+Thomae's function but this didn't seem to work either.
I was also wondering what would happen if just one of the two function is Riemann Integrable, the strict inequality holds or we'd obtain something even better?
Any help,hint or solution would be appreciated.
Let $d$ be Dirichlet's function. Then $$f=d\quad g=1-d$$ is a simple counterexample as all three integrals evaluate to $1$.