On Completely Continuous Integration Operators of a Vector Measure

Let

be a vector measure taking values in a Banach space

X X

. We prove that if the integration operator

I_m: L^1(m) \to X I m : L 1 ( m ) → X

,

I_m(f)=\int f \, dm I m ( f ) = ∫ f   d m

, is completely continuous and

X X

is Asplund, then

m m

has finite variation and

L^1(m) =L^1(|m|) L 1 ( m ) = L 1 ( ∣ m ∣ )

.