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  "body": "\n## Definition\n\nLet $\\mathcal{F}$ be an [[algebra of sets|PFMGZ7988V]] on a set $\\Omega$, and let $B$ be a [[Banach space|06GZ956657]].\nA *vector measure* is a map $F \\colon \\mathcal{F} \\to B$ that is finitely additive: $F(E_1 \\cup E_2) = F(E_1) + F(E_2)$ for all disjoint $E_1, E_2 \\in \\mathcal{F}$.\nA vector measure $F$ is *countably additive* if, for every sequence $(E_n)$ of pairwise disjoint sets in $\\mathcal{F}$ whose union $E$ lies in $\\mathcal{F}$,\n$$\\lim_{N \\to \\infty} \\Big\\| F(E) - \\sum_{n=1}^{N} F(E_n) \\Big\\| = 0.$$\n\n## Remarks\n\nExample: let $\\mathcal{F}$ be the Lebesgue measurable subsets of $[0, 1]$, let $1 \\le p \u003c \\infty$, and let $F(E) = \\mathbf{1}_E \\in L^p([0, 1])$.\nThen $F$ is a countably additive vector measure.\nFinite additivity holds because $\\mathbf{1}_{E_1 \\cup E_2} = \\mathbf{1}_{E_1} + \\mathbf{1}_{E_2}$ for disjoint sets.\nFor a disjoint sequence, $\\norm{F(E) - \\sum_{n \\le N} F(E_n)}_p = \\lambda\\big( \\bigcup_{n \u003e N} E_n \\big)^{1/p} \\to 0$.\n\nThe same map with values in $L^\\infty([0, 1])$ is a vector measure that is not countably additive.\nFor $E_n = [2^{-n}, 2^{-n+1})$, the remainder $\\mathbf{1}_{(0, 2^{-N})}$ has $L^\\infty$ norm $1$ for every $N$.\n\nNon-example: $G(E) = \\lambda(E)^2 \\in \\R$ is not a vector measure, since $G([0, 1]) = 1$ but $G([0, \\tfrac12)) + G([\\tfrac12, 1]) = \\tfrac12$.\n\nA real or complex measure that is finite is a countably additive vector measure with $B = \\R$ or $B = \\C$.\n",
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