real analysis - Finite Measure Space integration

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space with finite $\mu$. Let $f,f_n:\Omega \to \overline{\mathbb{R}}$ be a $\mathcal{A}-$measurable function $(n \in \mathbb{N})$.

For every $\varepsilon >0$,

$$lim_{n \to \infty} \mu (\bigcup_{m \geq n} x \in \Omega:f_m(x) > f(x)+ \varepsilon))=0$$

it means we may choose an integer $N$ large enough so that

$$\mu (\bigcup_{m \geq N} \left \{ x \in \Omega:f_m(x) > f(x)+ \varepsilon)) \right \})<\delta?$$