Below we use Bochner measurability and Bochner integral. Let
- $(X, \mathcal A, \mu)$ be a complete finite measure space,
- $(E, | \cdot |)$ a Banach space,
- $S (X)$ the space of $\mu$-simple functions from $X$ to $E$, and
- $L^0 (X)$ the space of $\mu$-measurable functions from $X$ to $E$.
- $L^1 (X)$ the space of $\mu$-integrable functions from $X$ to $E$.
For simplicity, we write $$ \mu (|f - g| > \delta) := \mu (\{x \in X : |f (x) - g(x)| > \delta\}) \quad \forall \delta >0. $$
Notice that the completeness of $(X, \mathcal A, \mu)$ implies $\{x \in X : |f (x) - g(x)| > \delta\} \in \mathcal A$. We define $$ \hat \rho (f, g) := \inf_{\delta >0} \{ \mu (|f - g| > \delta) +\delta \} \quad \forall f, g \in L^0 (X). $$
Then $\hat \rho$ is a pseudometric on $L^0(X)$. Also, $\hat \rho (f_n, f) \to 0$ IFF $f_n \to f$ in measure, i.e., $$ \mu (\{x \in X : |f_n (x)-f(x)| > \varepsilon\}) \xrightarrow{n \to \infty} 0 \quad \forall \varepsilon>0, $$
Could you elaborate on the completeness of $(L^0 (X), \hat \rho)$?
Thank you so much for your help!
