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, notions, absolute and/or continuity:
“Hesitation increases in relation to risk in equal proportion to age.”
—Ernest Hemingway (1899–1961)
“Even the simple act that we call “going to visit a person of our acquaintance” is in part an intellectual act. We fill the physical appearance of the person we see with all the notions we have about him, and in the totality of our impressions about him, these notions play the most important role.”
—Marcel Proust (1871–1922)
“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)
“Only the family, society’s smallest unit, can change and yet maintain enough continuity to rear children who will not be “strangers in a strange land,” who will be rooted firmly enough to grow and adapt.”
—Salvador Minuchin (20th century)