Representing Yosida-Hewitt decompositions for classical and non-commutative vector measures

Let m be a bounded, real valued measure on a field of sets. Then, by the Yosida-Hewitt theorem, m has a unique decomposition into the sum of a countably additive and a singular measure. We show here that, in contrast to the classical arguments, this decomposition can be achieved by constructing the countably additive component. From this we obtain a simple formula for the countably additive part of a (strongly bounded) vector measure. We develop these ideas further by considering a weakly compact operator T on a von Neumann algebra M. It turns out that T has a unique decomposition into T_N +T_S, where T_S is singular, T_N is completely additive on projections and, for each x in M, there exists an increasing sequence of projections (p_n)(n = 1,2…), such that

T_N(x)=limT(p_nxp_n).

When M has a faithful representation on a separable Hilbert space, then we can fix a sequence of projections (p_n)(n = 1,2…) such that the above equation holds for every choice of x in M. For general M, there exists an increasing net of projections < q_F > such that, for every y in M,

lim_FT_N(y)−T(q_Fyq_F)=0.