Relation Between The Two Notions of Absolute Continuity
A finite measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function
is locally an absolutely continuous real function. In other words, a function is locally absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.
If the absolute continuity holds then the Radon-Nikodym derivative of μ is equal almost everywhere to the derivative of F.
More generally, the measure μ is assumed to be locally finite (rather than finite) and F(x) is defined as μ((0,x]) for x>0, 0 for x=0, and -μ((x,0]) for x<0. In this case μ is the Lebesgue-Stieltjes measure generated by F. The relation between the two notions of absolute continuity still holds.
Read more about this topic: Absolute Continuity
Famous quotes containing the words relation between, relation, notions, absolute and/or continuity:
“There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
—Blaise Pascal (1623–1662)
“Light is meaningful only in relation to darkness, and truth presupposes error. It is these mingled opposites which people our life, which make it pungent, intoxicating. We only exist in terms of this conflict, in the zone where black and white clash.”
—Louis Aragon (1897–1982)
“The herd of mankind can hardly be said to think; their notions are almost all adoptive; and, in general, I believe it is better that it should be so; as such common prejudices contribute more to order and quiet, than their own separate reasonings would do, uncultivated and unimproved as they are.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“My mother and father are the only people on the whole planet for whom I will never begrudge a thing. Should I achieve great things, it is the work of their hands; they are splendid people and their absolute love of their children places them above the highest praise. It cloaks all of their shortcomings, shortcomings that may have resulted from a difficult life.”
—Anton Pavlovich Chekhov (1860–1904)
“The dialectic between change and continuity is a painful but deeply instructive one, in personal life as in the life of a people. To “see the light” too often has meant rejecting the treasures found in darkness.”
—Adrienne Rich (b. 1929)