Is the following proposition true?
Let $f(x)$ be a real-valued function defined on $[a,b] \subset \mathbb{R}$, and suppose that the integral, $$ I = \int_a^b f(x) dx, $$ exists in the sense of Riemann integral. If $0< |I| < \infty$, then $$ 0< \frac{1}{|I|}\int_a^b |f(x)| dx<\infty. $$
The absolute value of a Riemann integrable function is Riemann integrable, so $\int_a^b |f(x)| \ dx < \infty$. In addition, you must have $0<\int_a^b |f(x)| \ dx$, for if the integral is zero then $|f(x)|=0$ almost everywhere on $[a,b]$, which would imply $\int_a^b f(x) \ dx = 0$ (i.e., $|I|=0$). Since $0<|I|<\infty$ as well, you do indeed get the result that $0<\frac{1}{|I|}\int_a^b|f(x)| \ dx <\infty$.
