$\int \left| f_n \right|< M$ then $\left| f_n(x) \right|\leq g(x)$

Question

Is it true that if for all $n \in \mathbb{N}$ : $\int_I \left| f_n(x) \right| \,\mathrm{d}x < M$, where $M \in \mathbb{R}$, then there exists an integrable function $g$ (in the Riemann sens) such that for all : $n$ : $\left| f_n(x) \right| \leq g(x)$ ?

I think there is a link with the dominated convergence theorem, but I am unable to find a counterexample.

Thank you.

Answer 1

Hint:

Think about a series of step functions that have shrinking width and increasing height.