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:
“Trying to love your children equally is a losing battle. Your children’s scorecards will never match your own. No matter how meticulously you measure and mete out your love and attention, and material gifts, it will never feel truly equal to your children. . . . Your children will need different things at different times, and true equality won’t really serve their different needs very well, anyway.”
—Marianne E. Neifert (20th century)
“No one thinks anything silly is suitable when they are an adolescent. Such an enormous share of their own behavior is silly that they lose all proper perspective on silliness, like a baker who is nauseated by the sight of his own eclairs. This provides another good argument for the emerging theory that the best use of cryogenics is to freeze all human beings when they are between the ages of twelve and nineteen.”
—Anna Quindlen (20th century)
“My beloved spake, and said unto me, Rise up, my love, my fair one,
and come away.
For lo, the winter is past, the rain is over and gone;
The flowers appear on the earth; the time of the singing of birds is come, and the voice of the turtle is heard in our land;”
—Bible: Hebrew The Song of Solomon (l. II, 10–12)
“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)