Corollary

— There exists a projector p of the space $\mathscr{M}_\mathbf{C}(T)$ onto the space $L^1_{\text{loc}}(T, \mu; \mathbf{C})$, whose kernel $\overline{p}^{-1}(0)$ is the set of complex measures alien to $\mu$, such that

for every complex measure $\theta$.

If $p$ is restricted to the set of bounded measures, one obtains a projector $p^1$ of the space $\mathscr{M}_\mathbf{C}^1(T)$ onto the space $L^1_\mathbf{C}(T, \mu)$; the relation $\| \theta \| = |\theta|(1)$ implies that $\| \theta \| = \| p^1(\theta) \| + \| \theta - p^1(\theta) \|$ for every bounded complex measure $\theta$.