Any example for lower integral $\underline{I}_a^b (s)+\underline{I}_a^b (t)\ne \underline{I}_a^b (s+t) $ for two bounded functions $s$ and $t$?

Question

Any example for $\underline{I}_a^b (s)+\underline{I}_a^b (t)\ne \underline{I}_a^b (s+t) $ for two bounded functions $s$ and $t$?

Since it is true that $L_f(M)+L_g(M)\le L_{f+g}(M) $ for partitions M, we find $\underline{I}_a^b (s)+\underline{I}_a^b (t)\lt \underline{I}_a^b (s+t) $. But I can't think of any real, concrete examples. Could someone help?

Answer 1

Say $s(x)=1$ for rational $x$, $s(x)=0$ for irrational $x$. Let $t=1-s$. Then the lower integrals of $s$ and $t$ are both zero.