ISSN:
1420-8903
Keywords:
Primary 39A20, 39A40, 20K99
;
Secondary 06A60
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary Thepositive half A + of an ordered abelian groupA is the set {x ∈ A∣x ⩾ 0} andM $$ \subseteq$$ A + is amodule if for allx, y ∈ M alsox + y, |x − y| ∈M. Ifθ ∈ A + \M thenM(θ) is the module generated byM andθ. S $$ \subseteq$$ M isunbounded inM if(∀x ∈ M)(∃y ∈ S)(x ⩽ y) and isdense inM if (∀x1, x2 ∈M)(∃y ∈ S) (x1 〈 x2 ℂ x1 ⩾ y ⩾ x2). IfM is a module, or a subgroup of any abelian group, a real-valuedg: M → R issubadditive ifg(x + y) ⩽ g(x) + g(y) for allx, y ∈ M. The following hold: (1) IfM andM * are modules inA andM $$ \subseteq$$ M * $$ \subseteq$$ A + then a subadditiveg:M → R can always be extended to a subadditive functionF:M * → R when card(M) = ℵ0 and card(M * ) ⩽ ℵ1, or wheneverM * possesses a countable dense subset. (2) IfZ $$ \subseteq$$ A is a subgroup (whereZ denotes the integers) andg:Z + → R is subadditive with $$\aleph$$ g(n)/n = − ∞ theng cannot be subadditively extended toA + whenA does not contain an unbounded subset of cardinality $$ \le 2^{\aleph _o }$$ . (3) Assuming the Continuum Hypothesis, there is an ordered abelian groupA of cardinality ℵ1 with a moduleM and elementθ∈A + /M for whichA + = M(θ), and a subadditiveg:M → R which does not extend toA +. This even happens withg ⩾ 0. (4) Letg:A + → R be subadditive on the positive halfA + ofA. Then the necessary and sufficient condition forg to admit a subadditive extension to the whole groupA is: sup{g(x + y) − g(x)∣x⩾ −y} 〈 + ∞ for eachy 〈 0 inA. (5) IfM is a subgroup of any abelian groupA andg:M → K is subadditive, whereK is an ordered abelian group, theng admits a subadditive extensionF:A → K. (6) IfA is any abelian group andg:A → R is subadditive, theng = λ + ϕ whereλ:A → R is additive andϕ ⩾ 0 is a non-negative subadditive functionϕ:A → R. IfA is aQ-vector spaceλ may be takenQ-linear. (7) Ifg:V → R is a continuous subadditive function on the real topological linear spaceV then there exists a continuous linear functionalλ:V → R and a continuous subadditiveϕ:V → R such thatg = λ + ϕ andϕ ⩾ 0. ifV = R n this holds for measurable subadditiveg with a continuousλ and measurableϕ.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01837942
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