Let $f:\mathbb{R}\to\mathbb{R}$ be a Riemann integrable probability density function on every closed interval of $\mathbb{R}$. As I explained in my previous question (Does there exist any probability density function $f:\mathbb{R}\to\mathbb{R}$ which is not Riemann integrable?) , Now I have 2 questions as follows:
Question(1) Is $\int_{-\infty}^{+\infty}f(x)dx=\int_{-\infty}^{0}f(x)dx+\int_{0}^{+\infty}f(x)dx$ convergent?
Question(2) Is $\lim_{x\to +\infty}f(x)=0$ true?
For question 1) the answer is YES and it follows from the fact that the Riemann integral of a Riemann integrable fucntion coincides with the Lebesgue integral. For question 2) the answer is NO: let $f(x)=c\sum nI_{(n,n+\frac 1 {n^{3}})}$ where $c$ is chosen such that $f$ integrates to $1$.