Let $X \in \mathbb{R}$ be a continuous random variable with density function $f$ (i.e. $f(x)\geq 0$ and $\int f(x)dx=1$). Does this mean that $\int |f(x)|dx < \infty$ i.e. $f \in L^1$?
(The reason I ask this question is because I know the characteristic function of $X$ always exists and I wanted to check that this is because $f \in L^1$.)
The probability density function is non-negative. Hence, $|f(x)|=f(x)$ and $$ \int |f(x)|\mathrm dx=\int f(x)\mathrm dx=1. $$ It follows that $f\in L^1$.