Let $f$ and $g$ be bounded on $[a , b]$ and $g \in \mathcal{R}(\alpha)$ on $[a ,b]$. Also $P$ is an arbitrary partition. If $f\le g$ and $\alpha$ is increasing on $[a , b]$ determine whether the following propositions are true or false:
$1.$ $L(P,f,\alpha) \le \int_a^b gd\alpha$
$2.$ $U(P,f,\alpha) \le \int_a^b gd\alpha$
My try: I think the first one is true since $$f\le g \implies \int_a^b fd\alpha \le \int_a^b gd\alpha \\ L(P,f,\alpha)\le \int_a^b fd\alpha \implies L(P,f,\alpha)\le \int_a^b gd\alpha$$ The second proposition is wrong. Let $f=g =x$ and $a = 0,b =2, P = \{0 ,1,2\}$. Then $U(P,f,\alpha) = 3$ and $\int_a^b gd\alpha = 2$. Clearly $3 \not \le 2$, so this is false.
Are my answers correct and precise?
For (1) your approach is not adequate, as it is possible that $f \leqslant g$ but $f \not\in \mathcal{R}(\alpha)$. This occurs when $f$ and $\alpha$ are both discontinuous at a point in $[a,b]$ from the right or left.
A simple argument is
$$L(P,f,\alpha) = \sum_{j=1}^n \inf_{x \in [x_{j-1},x_j]}f(x) \cdot (\alpha(x_j) - \alpha(x_{j-1})) \leqslant \sum_{j=1}^n \inf_{x \in [x_{j-1},x_j]}g(x) \cdot (\alpha(x_j) - \alpha(x_{j-1}))\\ \leqslant \int_a^b g \, d\alpha $$