Originally Posted by
Laurent What do you mean?
In general, functions $\displaystyle f\geq 0$ such that $\displaystyle \int f<\infty$ are not non-increasing. Some of them converge to 0, others don't.
Here's an example. Let me describe the graph of $\displaystyle f$. It consists of triangular peaks that get thinner and higher : around $\displaystyle n$, we put a triangular peak of width $\displaystyle \frac{1}{n^3}$ and height $\displaystyle n$. (Is it clear?)
The area of the triangle at $\displaystyle n$ is $\displaystyle \frac{1}{n^2}$. Since $\displaystyle \sum_n \frac{1}{n^2}<\infty$, the total area under the curve is finite, i.e. $\displaystyle \int f(x)dx<\infty$. However, $\displaystyle f$ clearly does not converge to 0. We even have $\displaystyle f(n)=n\to +\infty$ for $\displaystyle n\in\mathbb{N},n\to\infty$.