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:
“This entire most beautiful order of good things is going to pass away after its measure has been exhausted; for both morning and evening were made in them.”
—St. Augustine (354–430)
“The theory of truth is a series of truisms.”
—J.L. (John Langshaw)
“If a man, notoriously and designedly, insults and affronts you, knock him down; but if he only injures you, your best revenge is to be extremely civil to him in your outward behaviour, though at the same time you counterwork him, and return him the compliment, perhaps with interest.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“As deaths have accumulated I have begun to think of life and death as a set of balance scales. When one is young, the scale is heavily tipped toward the living. With the first death, the first consciousness of death, the counter scale begins to fall. Death by death, the scales shift weight until what was unthinkable becomes merely a matter of gravity and the fall into death becomes an easy step.”
—Alison Hawthorne Deming (b. 1946)