A signed measure completeness criterion

1989 ◽  
Vol 17 (3) ◽  
pp. 253-261 ◽  
Author(s):  
Anatolij Dvurečenskij ◽  
Sylvia Pulmannová
Keyword(s):  
Signed Measure ◽  
Completeness Criterion

On (L1)* for General Measure Spaces

1963 ◽  
Vol 6 (2) ◽  
pp. 211-229 ◽  
Author(s):  
H. W. Ellis ◽  
D. O. Snow
Keyword(s):  
Measurable Function ◽  
Measure Space ◽  
Finite Measure ◽  
Signed Measure ◽  
General Measure ◽  
Measure Spaces ◽  
Arbitrary Measure ◽  
Image Position ◽  
Nikodym Theorem

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

Measure generation by Euler functionals

1995 ◽  
Vol 27 (03) ◽  
pp. 606-626
Author(s):  
R. V. Ambartzumian
Keyword(s):  
Sufficient Conditions ◽  
Additive Functional ◽  
Signed Measure ◽  
Convex Polygons ◽  
Bounded Convex

Guided by analogy with Euler's spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ2(the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed measures in the space of lines in ℝ2. In this way we obtain two sets of conditions which ensure that a segment function corresponds to a signed measure in the space of lines. The latter conditions are also necessary.

scholarly journals A decomposition of additive functionals of finite energy

1979 ◽  
Vol 74 ◽  
pp. 137-168 ◽  
Author(s):  
Masatoshi Fukushima
Keyword(s):  
Brownian Motion ◽  
Sobolev Space ◽  
Bounded Variation ◽  
Finite Energy ◽  
Signed Measure ◽  
Ito Formula ◽  
Tempered Distribution ◽  
Additive Functionals ◽  
Right Hand ◽  
The Right

The celebrated Ito formula for the n-dimensional Brownian motion Xt and for u ∈ C2(Rn) runs as follows:(0.1) In § 6 of this paper we extend this to the case where u is any element of the Sobolev space H1R(n) and accordingly Δu is a tempered distribution which is not even a signed measure in general. As a consequence the second term of the right hand side of (0.1) may not be of bounded variation in t.

Hyperbolic Valued Signed Measure

2018 ◽  
Vol 55 (7) ◽  
pp. 515-522 ◽  
Author(s):  
Chinmay Ghosh ◽  
Sanjoy Biswas ◽  
Taha Yasin
Keyword(s):  
Signed Measure

Some orthogonal and complete sets of Bessel functions associated with the vibrating plate

1982 ◽  
Vol 91 (3) ◽  
pp. 503-515 ◽  
Author(s):  
J. R. Higgins
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Main Concern ◽  
Edge Type ◽  
Completeness Criterion ◽  
Complete Sets ◽  
Clamped Edge ◽  
Vibrating Plate

AbstarctSome orthogonal sets of Bessel functions of real order v are identified using the equation Δ2u = utt of the vibrating plate. Our main concern is with the L2 completeness of such sets, and we prove that the well known ‘clamped edge’ type is complete for v > -1, thus completing a result of E. Dahlberg. We also study a very closely related set and show that it needs an extra (non-Bessel) function for completeness.Our method for proving the completeness is based on one given by H. Hochstadt in connection with Dini functions. We have found it necessary to reorganize Hoch-stadt's method and correct some errors contained in it.Certain isolated values of v require special attention and we treat these by subjecting the Dalzell completeness criterion to a continuity argument.

scholarly journals ENTROPY AND RENORMALIZED SOLUTIONS FOR THE p(x)-LAPLACIAN EQUATION WITH MEASURE DATA

2010 ◽  
Vol 82 (3) ◽  
pp. 459-479 ◽  
Author(s):  
CHAO ZHANG ◽  
SHULIN ZHOU
Keyword(s):  
Existence And Uniqueness ◽  
Signed Measure ◽  
Entropy Solutions ◽  
Measure Data ◽  
Variable Exponents ◽  
Renormalized Solutions ◽  
Laplacian Equation

AbstractIn this paper we prove the existence and uniqueness of both entropy solutions and renormalized solutions for the p(x)-Laplacian equation with variable exponents and a signed measure in L1(Ω)+W−1,p′(⋅)(Ω). Moreover, we obtain the equivalence of entropy solutions and renormalized solutions.

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