Prove that $\mathcal{BFA}(X, \mathfrak{M}, \mu)$ is a linear space on which $\|.\|_{var}$ is a norm.

Question

Show that $\mathcal{BFA}(X, \mathfrak{M}, \mu)$ is a linear space on which $\|.\|_{var}$ is a norm. then show that this normed linear space is a Banach space.

Any hints for the triangle inequality proof and the completeness proof ?

Answer 1

Let $\nu, \omega \in \mathcal{BAF}(X, \mathfrak{M}, \mu)$. Fix a disjoint, finite collection of measurable sets $\{E_k\}_{k=1}^n$ and show that $$ \sum_{k=1}^n|[\nu+\omega](E_k)| \leq \sum_{k=1}^n |\nu(E_k)| + \sum_{k=1}^n|\omega(E_k)| $$ Taking the supremum on either side will give you the desired inequality.