$f,g \in {R}(\alpha)$ riemann integrals, on $[a,b]$ which is increasing in $\mathbb{R}$. prove if $\int_{[a,b]} |f|^pd\alpha$=0 then $\int_{[a,b]} fgd\alpha=0$.
my attempt: since I can't assume f is continuous, and f and g are Riemann integrable I can write $|f|^p$ and $fg$ are Riemann integrable.
\begin{equation} |\sum^n_{i=1}|f(t_i)|^p\Delta\alpha_i -\int^b_a |f|^pd\alpha|<\epsilon \end{equation} and
\begin{equation} |\sum^n_{i=1}f(t_i)g(t_i)\Delta\alpha_i -\int^b_a fgd\alpha|<\epsilon_2 \end{equation}
\begin{equation} |\sum^n_{i=1}|f(t_i)|^p\Delta\alpha_i -\int^b_a |f|^pd\alpha|<\epsilon_1 \end{equation} since $\int^b_a |f|^pd\alpha=0$
\begin{equation} |\sum^n_{i=1}|f(t_i)|^p\Delta\alpha_i|<\epsilon_1 \end{equation}
consider \begin{equation} |\sum^n_{i=1}f(t_i)g(t_i)\Delta\alpha_i|=|\sum^n_{i=1}f(t_i)\alpha_i^{1/p}g(t_i)\Delta\alpha_i^{1/q}| \end{equation} where $1/p+1/q=1$.
with holders inequality
\begin{equation} |\sum^n_{i=1}f(t_i)\alpha_i^{1/p}g(t_i)\Delta\alpha_i^{1/q}| \leq \big(\sum^n_{i=1}|f(t_i)|^p\Delta\alpha_i\big)\big(\sum^n_{i=1}|g(t_i)|^q\Delta\alpha_i\big) \end{equation}
\begin{equation} |\sum^n_{i=1}f(t_i)g(t_i)\Delta\alpha_i| \leq \big(\sum^n_{i=1}|f(t_i)|^p\Delta\alpha_i\big)\big(\sum^n_{i=1}|g(t_i)|^q\Delta\alpha_i\big) \end{equation}
\begin{equation} |\sum^n_{i=1}f(t_i)g(t_i)\Delta\alpha_i| \leq \epsilon_1 \big(\sum^n_{i=1}|g(t_i)|^q\Delta\alpha_i\big) \end{equation}
\begin{equation} |\int^b_a fgd\alpha|\leq|\sum^n_{i=1}f(t_i)g(t_i)\Delta\alpha_i|+\epsilon_2 \end{equation}
\begin{equation} |\int^b_a fgd\alpha|\leq \epsilon_1 \big(\sum^n_{i=1}|g(t_i)|^q\Delta\alpha_i\big)+\epsilon_2 \end{equation}
\begin{equation} |\int^b_a fgd\alpha|\leq \epsilon \end{equation}
after this point, I'm not sure how I can go further, if you have any ideas appreciate your help.