What is caratheodory condition?
In mathematics, Carathéodory’s existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano’s existence theorem.
What is caratheodory function?
Definition 1.1. Let X, Y be topological spaces and (T, M,µ) be a measur- able space. We say that f : T × X → Y is a Carathéodory function if. (i) f (·,u) is measurable for each u, (ii) f (t, ·) is continuous for each t.
How do you find the outer measure?
Definition of a regular outer measure
- for any subset A of X and any positive number ε, there exists a μ-measurable subset B of X which contains A and with μ(B) < μ(A) + ε.
- for any subset A of X, there exists a μ-measurable subset B of X which contains A and such that μ(B) = μ(A).
What is the importance of the existence theorem?
A theorem stating the existence of an object, such as the solution to a problem or equation. Strictly speaking, it need not tell how many such objects there are, nor give hints on how to find them.
What is uniqueness theorem in statistics?
A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model).
What is measure and integration?
Measure and Integration. then-dimensional volume of the set where the function is greater than some number y. This volume is a well-behaved, monotone nonincreasing function of the number y, which then can be integrated in the manner of Riemann.
What is Lebesgue measure and integration?
The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure.
What is almost everywhere in measure theory?
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities.
Is Lebesgue measure an outer measure?
A set Z is said to be of (Lebesgue) measure zero it its Lebesgue outer measure is zero, i.e. if it can be covered by a countable union of (open) intervals whose total length can be made as small as we like. If Z is any set of measure zero, then m∗(A ∪ Z) = m∗(A). The outer measure of a finite interval is its length.
What is existence and uniqueness theorem?
Existence and Uniqueness Theorem (EUT) If f, ∂ f ∂ y , and ∂ f ∂ y ˙ are continuous in a closed box B in three-dimensional space (t-y- space) and the point ( t 0 , y 0 , y ˙ 0 ) lies inside B, then the IVP has a unique solution on some t-interval I containing .
What is stated by the existence theorem?
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase “there exist(s)”, or it might be a universal statement whose last quantifier is existential (e.g., “for all x, y, there exist(s) …”).
What is Carathéodory’s extension theorem?
In measure theory, Carathéodory’s extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring R of subsets of a given set Ω can be extended to a measure on the σ-algebra generated by R, and this extension is unique if the pre-measure is σ-finite.
What is the Carathéodory construction of measure space?
\\mu:\\Sigma o [0,\\infty] μ: Σ → [0,∞], is known as the Carathéodory construction. \\Sigma Σ. Note that by monotonicity, any such subset is also \\mu μ -null. Completeness implies certain desirable properties of the measure space, such as the following.
What is the significance of the sigma additivity theorem?
This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending