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:
“What cannot stand must fall; and the measure of our sincerity and therefore of the respect of men, is the amount of health and wealth we will hazard in the defence of our right. An old farmer, my neighbor across the fence, when I ask him if he is not going to town-meeting, says: “No, ‘t is no use balloting, for it will not stay; but what you do with the gun will stay so.””
—Ralph Waldo Emerson (1803–1882)
“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)
“When the landscape buckles and jerks around, when a dust column of debris rises from the collapse of a block of buildings on bodies that could have been your own, when the staves of history fall awry and the barrel of time bursts apart, some turn to prayer, some to poetry: words in the memory, a stained book carried close to the body, the notebook scribbled by hand—a center of gravity.”
—Adrienne Rich (b. 1929)
“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)