Integral Form With Locally Finite Measures
Let I denote an interval of the real line of the form or ) < ∞ for all t ∈ I (this is certainly satisfied when μ is a locally finite measure). Assume that u is integrable with respect to μ in the sense that
and that u satisfies the integral inequality
If, in addition,
- the function α is non-negative or
- the function t → μ is continuous for t ∈ I and the function α is integrable with respect to μ in the sense that
then u satisfies Grönwall's inequality
for all t ∈ I, where Is,t denotes to open interval (s, t).
Read more about this topic: Gronwall's Inequality
Famous quotes containing the words integral, form, locally, finite and/or measures:
“... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
—Angelina Grimké (1805–1879)
“As for the terms good and bad, they indicate no positive quality in things regarded in themselves, but are merely modes of thinking, or notions which we form from the comparison of things with one another. Thus one and the same thing can be at the same time good, bad, and indifferent. For instance music is good for him that is melancholy, bad for him who mourns; for him who is deaf, it is neither good nor bad.”
—Baruch (Benedict)
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)
“Are not all finite beings better pleased with motions relative than absolute?”
—Henry David Thoreau (1817–1862)
“A lake is the landscape’s most beautiful and expressive feature. It is earth’s eye; looking into which the beholder measures the depth of his own nature. The fluviatile trees next the shore are the slender eyelashes which fringe it, and the wooded hills and cliffs around are its overhanging brows.”
—Henry David Thoreau (1817–1862)