I'm trying to prove that for any bounded functions $f,g : [a,b] \to \mathbb{R}$ the lower Riemann integrals obey the inequality
$$\underline{\int}_a^b(f(x)+g(x))dx \leq \underline{\int}_a^bf(x)dx + \underline{\int}_a^bg(x)dx $$
Using $\inf f(x) + \inf g(x) \leq \inf (f(x) + g(x))$, I was able to show that for lower sums with a partition $P$
$$ L(P,f) + L(P,g)\leq L(P,f+g)$$
I know that the lower integral is the supremum of lower sums for all partitions so
$$\sup_{P}(L(P,f) + L(P,g))\leq \sup_{P}L(P,f+g)=\underline{\int}_a^b(f(x)+g(x))dx $$
Also since $L(P,f) + L(P,g) \leq \sup_{P}L(P,f) + \sup_{P}L(P,g)$ we know that
$$\sup_{P}(L(P,f) + L(P,g))\leq \sup_{P}L(P,f) + \sup_{P}L(P,g)= \underline{\int}_a^bf(x)dx + \underline{\int}_a^bg(x)dx$$
But this does not help me show the desired inequality.
Can anyone please help me continue?
Best Answer
Your inequality for the lower sums is correct. However, your upper bounds, while both correct, do not relate the lower integrals.
Assume on the contrary that
$$\underline{\int}_a^b [f(x)+g(x)] \, dx < \underline{\int}_a^b f(x) \, dx + \underline{\int}_a^b g(x) \,dx .$$ Then $$\underline{\int}_a^b [f(x)+g(x)] \, dx - \underline{\int}_a^b g(x) \,dx < \underline{\int}_a^b f(x) \, dx,$$ and there exists a partition $P$ such that $$\underline{\int}_a^b [f(x)+g(x)] \, dx - \underline{\int}_a^b g(x) \,dx < L(P,f) \leqslant \underline{\int}_a^b f(x) \, dx.$$
Hence,
$$\underline{\int}_a^b [f(x)+g(x)] \, dx - L(P,f) < \underline{\int}_a^b g(x) \, dx,$$ and there exists a partition $P’$ such that $$\underline{\int}_a^b [f(x)+g(x)] \, dx - L(P,f) < L(P’,g) \leqslant \underline{\int}_a^b g(x) \, dx,$$ and $$\underline{\int}_a^b [f(x)+g(x)] \, dx < L(P,f) + L(P’,g) .$$
Now take a common refinement of the partitions $Q = P \cup P'$. Lower sums increase as partitions are refined and we have $L(Q,f) \geqslant L(P,f)$ and $L(Q,g) \geqslant L(P’,g).$
It follows that
$$L(Q,f+g) \leqslant \underline{\int}_a^b [f(x)+g(x)] \, dx < L(P,f) + L(P’,g) \leqslant L(Q,f) + L(Q,g).$$
This contradicts the inequality for lower sums, and, therefore
$$\underline{\int}_a^b f(x) \, dx + \underline{\int}_a^b g(x) \, dx \leqslant \underline{\int}_a^b [f(x) + g(x)] \, dx. $$
Note that if $f$ and $g$ were Riemann integrable, you would have a strict equality.