We define the semivariation of a vector measure $\mu:\mathcal{B}(\mathbb{T})\to E$ as

$$\|{\mu}\|(A)=\sup\lbrace{|\langle e^*,\mu\rangle|(A),\; e^*\in E^*,\;\|e^*\|=1\rbrace}$$

where $\langle e^*,\mu\rangle(A)=e^*(\mu(A))$, and $|\cdot|$ is the total variation for scalar measures.

I have read that in the case in which $E$ is a dual space, $E=F^*$, one can obtain this semivariation also in the following manner:

$$\|{\mu}\|(A)=\sup\lbrace{|\langle \mu,f\rangle|(A),\; f\in F,\;\|f\|=1\rbrace}$$

where $\langle \mu,f\rangle(A)=\mu(A)(f)$, and $|\cdot|$ is again the total variation for scalar measures.

How can one go from one definition to the other? Is it related with Hahn-Banach theorem?

  • $\begingroup$ Wouldn't this follow from Goldstine's Theorem (A normed linear space $X$ is weak* dense in $X^{**}$)? $\endgroup$ – David Mitra Apr 15 at 11:13

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