Is it true that $L_{f+g}(M)\le L_f(M)+L_g(M)$ if $M$ is a partition on [a,b] and $f, g$ are two bounded functions?

Question

Here I have a bit confusion, if $f$ is positive and $g$ is negative, does that mean the total lower sum is $L_f - |L_g|$ or is it the absolute value?

If $f$ and $g$ are both positive, how do you know that $L_{f+g}$ is exactly the sum of $L_f(M)$ and $L_g(M)$?

In addition, what do you need to formally prove this? Any help will be great.

Answer 1

The signs of $f(x)$ and $g(x)$ are not relevant. Also the lower sum of the sum of functions does not necessarily equal the sum of the lower sums. This only applies to Riemann sums with the same tags.

Note that on any sub-interval $I$ of the partition and $x \in I$,

$$\inf_I f(x) \leqslant f(x),\\ \inf_I g(x) \leqslant g(x).$$

Hence, for every $x \in I$

$$\inf_I f(x)+\inf_I g(x) \leqslant f(x)+ g(x),$$

and

$$\inf_I f(x)+\inf_I g(x) \leqslant \inf_I[f(x)+ g(x)].$$

So it appears the inequality should be reversed:

$$L_f(M) + L_g(M) = \sum_{k=1}^n \inf_{[x_{k-1},x_k]}f(x)(x_k - x_{k-1}) + \sum_{k=1}^n \inf_{[x_{k-1},x_k]}g(x)(x_k - x_{k-1}) \\ \leqslant \sum_{k=1}^n \inf_{[x_{k-1},x_k]}[f(x)+g(x)](x_k - x_{k-1}) = L_{f+g}(M)$$