Let $(X, \mathcal X, \mu )$ be a measure space, and let $ L(X)$ be the space of measurable functions $f: X \to \mathbb C$. Show that the sets $B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) - g(x) | \ge r \} < \epsilon ) \}$ for $ f \in L(X), \epsilon > 0, r> 0$ form the base for a topology that turns $L(X)$ into a topological vector space, and that a sequence $f_n \in L (X)$ converges to a limit $f$ in this topology if and only if it converges in measure.
I stuck at the begining, I try to show those sets form a base by showing for any $ g \in B(f, \epsilon , r ) \cap B(f', \epsilon ' ,r ')$, there exists a $g \in B( h, \epsilon_0, r_0) \subset B(f, \epsilon , r ) \cap B(f', \epsilon ' ,r ')$. But I failed to achieve this. Any help is appreciated.