What is convergence in measure theory?

What is convergence in measure theory?

In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence in measure, consider a sequence of measures μn on a space, sharing a common collection of measurable sets.

What is strong convergence?

Strong convergence is the type of convergence usually associated with convergence of a sequence. More formally, a sequence of vectors in a normed space (and, in particular, in an inner product space )is called convergent to a vector in if. SEE ALSO: Convergent Sequence, Inner Product Space, Weak Convergence.

Is convergence in probability weak convergence?

If the metric space X is compact, the Riesz representation theorem implies that Mb(X) is the dual of the space C(X) of continuous functions and hence the weak convergence of a sequence of probability measures {μn}⊂P(X) coincides with the weak∗ convergence.

What is a tight sequence?

Theorem 1.8 A sequence of probability measures {πn}∞ n=1 on (Dd,Dd) is tight if and only if. ∀T > 0,ϵ> 0 there exists Kϵ,δϵ > 0 such that: (i) lim supn→∞ πn(x ∈ Dd : ||x||T ≥ Kϵ) < ϵ

Does convergence in measure imply pointwise convergence?

Although convergence in measure does not imply pointwise convergence, we do have the following weaker (but still very useful) conclusion. If fn → f in L1(X), then there exists a subsequence {fnk }k∈N such that fnk → f pointwise a.e.

What is meant by weak convergence?

A sequence of vectors in an inner product space is called weakly convergent to a vector in if. Every strongly convergent sequence is also weakly convergent (but the opposite does not usually hold). This can be seen as follows. Consider the sequence that converges strongly to , i.e., as .

How do you show weak convergence?

IF space X is reflexive, then we can replace x ∈ X∗ with x ∈ X to show that weak* convergence implies weak convergence. Therefore weak and weak* convergence are equivalent on reflexive Banach spaces.

What is square limit?

We say that is mean-square convergent (or convergent in mean-square) if and only if there exists a square integrable random variable such that. The variable is called the mean-square limit of the sequence and convergence is indicated by or by. The notation indicates that convergence is in the Lp space.

How do you measure tightness?

A family Γ of probability measures in P(S) is said to be tight if, given any ε > 0, there exists a compact set K = Kε of S such that: P(K) > 1 − ε, for all P in Γ.