Measure Theory On Time Scales
Associated with every time scale is a natural measure defined via
where denotes Lebesgue measure and is the backward shift operator defined on . The delta integral turns out to be the usual Lebesgue–Stieltjes integral with respect to this measure
and the delta derivative turns out to be the Radon–Nikodym derivative with respect to this measure
Read more about this topic: Time-scale Calculus
Famous quotes containing the words measure, theory, time and/or scales:
“Who can measure the advantages that would result if the magnificent abilities of these women could be devoted to the needs of government, society and home, instead of being consumed in the struggle to obtain their birthright of individual freedom? Until this be gained we can never know, we can not even prophesy the capacity and power of women for the uplifting of humanity.”
—Susan B. Anthony (1820–1906)
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)
“We are double-edged blades, and every time we whet our virtue the return stroke straps our vice. Where is the skillful swordsman who can give clean wounds, and not rip up his work with the other edge?”
—Henry David Thoreau (1817–1862)
“For these have governed in our lives,
And see how men have warred.
The Cross, the Crown, the Scales may all
As well have been the Sword.”
—Robert Frost (1874–1963)