A bounded sequence is not dominated by an integrable function on an infinite measure space

Let $(\Omega,\mathbf F,\mu)$ be a measure space and let $f_n:\Omega\to\bar{\mathbb{R}}$ be a sequence of extended Borel measurable functions. Suppose $|f_n|\le100$, $\forall n=1,2,\dots$, and $\lim_{n\to\infty}f_n=f$ a.e. $\mu$. Then, which of the following statements is false?