Sequence of measurable functions and limit

In summary: S= {x | lim f_n(x) exists}5. S is measurable since f is measurable.6. f^{-1}(E) is measurable since f is measurable.7. Hence, S is in \mathcal{A}.In summary, we can prove that the set of points where a sequence of measurable functions converges is also measurable.
  • #1
complexnumber
62
0

Homework Statement



Given a [tex]\sigma[/tex]-algebra [tex](X,\mathcal{A})[/tex], let [tex]f_n : X \to
[-\infty,\infty][/tex] be a sequence of measurable functions. Prove that
the set [tex]\{ x \in X | \lim f_n (x) \text{ exists} \}[/tex] is in
[tex]\mathcal{A}[/tex].

Homework Equations



Let [tex](X,\mathcal{A})[/tex] be a [tex]\sigma[/tex]-algebra and [tex]M_{+}(X) := [/tex] set
of all functions [tex]f : X \to [0,\infty][/tex] which are
[tex]\mathcal{A}[/tex]-measurable. Let [tex]f_n \in M_+(X)[/tex], then if [tex]\displaystyle \lim_{n \to \infty} f_n(x) \exists[/tex] [tex]\forall
x \in X[/tex], then [tex]\displaystyle f := \lim_{n \to \infty} f_n \in
M_+(X)[/tex].

Any [tex]f : X \to [-\infty,\infty][/tex] can be written [tex]f = f_+ - f_-[/tex]
where [tex]f_+(x) := \sup \{ f(x), 0 \}[/tex], [tex]f_-(x) := - \inf \{ f(x), 0
\}[/tex]. So [tex]f[/tex] is [tex]\mathcal{A}[/tex]-measurable iff [tex]f_+, f_- \in M_+(X)[/tex].

The Attempt at a Solution



Should I use the above two facts to show that there exists a measurable [tex]\displaystyle f = \lim_{n \to \infty} f_n[/tex] and hence anything [tex]f^{-1}(E) \in \mathcal{A}, E \in \mathbb{R}[/tex]?
 
Physics news on Phys.org
  • #2
Here's a sketch of a proof.
1. Define f(x) =lim f_n(x) if f_n(x) converges ,
= 0 otherwise.
2. Using Lebesgue's dominated convergence theorem, show that f is measurable.
3. If a measurable function exists on a set S, its characteristic function ,too, is measurable.
 

What is the definition of a sequence of measurable functions?

A sequence of measurable functions is a collection of functions that are defined on a common domain and range, and each function is measurable with respect to a given sigma-algebra.

How do you determine the limit of a sequence of measurable functions?

The limit of a sequence of measurable functions is determined by taking the pointwise limit of the functions at each point in the common domain. If the resulting function is also measurable, it is the limit of the sequence.

Why is the concept of a sequence of measurable functions important in mathematics?

The concept of a sequence of measurable functions is important because it allows for the study of convergence and continuity in more general settings. It is used in many areas of mathematics, including measure theory, functional analysis, and probability theory.

Can a sequence of measurable functions have more than one limit?

Yes, a sequence of measurable functions can have multiple limits. This can occur if the functions have different domains or if the limit of the sequence is not unique.

What is the relationship between convergence and measurability in a sequence of measurable functions?

Convergence and measurability are closely related in a sequence of measurable functions. If a sequence of measurable functions converges pointwise, the limiting function will also be measurable. Conversely, if a sequence of measurable functions does not converge pointwise, the limiting function will not be measurable.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
588
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
695
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
13
Views
635
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
658
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
942
Back
Top