On the generalized F. and M. Riesz theorem

The theorem of F. and M. Riesz

Suppose that G is a locally compact abelian group and that P ⊂ G is a semigroup (satisfies P + P ⊂ P ) that is closed and satisfies P ∩ (−P ) = {0} and P ∪ (−P ) = G. We define a total order on G by x ≤ y when y − x ∈ P . We verify that this is indeed a total order. (We remark that nowhere in this do we show the significance of P being closed; but in this note we shall be speaking about discret...

S ep 2 00 1 AN F . AND M . RIESZ THEOREM FOR PLANAR VECTOR FIELDS

We prove that solutions of the homogeneous equation Lu = 0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if Ω is an open subset of the plane with smooth boundary, u ∈ C 1 (Ω) satisfies Lu = 0 on Ω, has tempered growth at the boundary, and its weak boundary value is a measure µ, then µ is absolutely continuo...

The Riesz representation theorem

On the equivalence theorem in f(R)-type generalized gravity

We investigate whether the equivalence theorem in f(R)-type gravity is valid also in quantum theory. It is shown that, if the canonical quantization is assumed, the equivalence does not hold in quantum theory.

Notes on L(Ω,M, μ) and the Riesz–Fischer Theorem

Let (Ω, M, µ) be a measure space. Then for any two complex valued measurable functions f and g, we have the Cauchy-Schwarz inequality: 0.1 THEOREM (Cauchy-Schwarz inequality). Let (Ω, M, µ) be a measure space. Then for any two complex valued measurable functions f and g such that