Remarks on the semivariation of vector measures with respect to banach spaces.

Suppose that L-q(nu)(circle times) over cap Y-gamma q = L-q(nu,Y) and X (circle times) over cap L-Delta p(p)(mu) = L-P(mu,X). It is shown that any L-P(mu)-valued measure has finite L-2 (nu)-semivariation with respect to the tensor norm L-2(nu)(circle times) over cap L-Delta p(p)(mu) for 1 <= p < infinity and finite L-q (nu)-semivaxiation with respect to the tensor norm L-q (nu)(circle times) over cap L-gamma q(p)(mu) whenever either q = 2 and 1 <= p <= 2 or q > max{p, 2}. However there exist measures with infinite L-q-semivaxiation with respect to the tensor norm L-q (nu)(circle times) over cap L-gamma q(p)(mu) for any 1 <= q < 2. It is also shown that the measure m(A) = chi A has infinite Lq -semivaxiation with respect to the tensor norm L-q (nu)(circle times) over cap L-gamma q(p)(mu) if q < p.