Definitions of Measurable Functions

In summary, the two definitions of measurable functions, one based on inverse images of open sets and the other on inverse images of Borel sets, are equivalent. This can be shown using transfinite induction, but it is also apparent by the definition of the Borel sigma algebra.
  • #1
chingkui
181
2
I have seen a number of definitions of measurable functions, particularly, for the following two definitions, are they the same?
1) A function f:X->R is measurable iff for every open subset T of R, the inverse image of T is measurable.
2) A function f:X->R is measurable iff for every Borel set B of R, the inverse image of B is measurable.
Here, R is the real line.
I suppose they are the same, but I haven't been able to show they are equivalent (one direction is easy, but the other one, by assuming any open set has measurable inverse image and showing that the inverse image of all Borel set is measurable, doesn't seem to be that easy, since Borel sets in general cannot be easily written down in terms of open sets).
 
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  • #2
since Borel sets in general cannot be easily written down in terms of open sets

Sure they can -- they're made from applying finitely many σ-algebra operations to open sets. (that is, complement and countable union)
 
  • #3
Sure they can -- they're made from applying finitely many σ-algebra operations to open sets. (that is, complement and countable union)
Hi Hurkyl, I am under the impression that a Borel set might not be generated under finitely many σ-algebra operations to open sets. Borel σ-algebra is defined in all the books I read as "the smallest σ-algebra containing the open sets", it is not defined as a constructive process of σ-algebra operations, so, I am not even sure whether countablely many σ-algebra operations to open sets is enough to generate all the Borel sets. That's why I am not sure how to show the two definitions of measurable functions are the same. Are you sure they all can be formed by a finite process? Thanks.
 
  • #4
Is the set of inverse images of sets under f a [itex]\sigma[/itex]-algebra?
 
  • #5
Bah, you're right, I've made this mistake before, but I've fixed it before. :smile: I meant to say:

Any Borel set can be formed by γ applications of the σ-algebra operations, for some ordinal number γ. (For example, letting γ be any ordinal with cardinality greater than that of P(R) suffices)

More explicitly:

Let B(0) be the set of all open sets.

For any ordinal number α > 0:
Let C(α) be the union of all C(β) with β < α
Then, define B(α) to be the the set of all:
(1) things in C(α)
(2) complements of things in C(α)
(3) things that are countable unions of things in C(α)


You can prove there is an upper bound by some clever theorem of set theory that I don't remember, or simply noting that once you've stopped adding sets, you have the class of Borel sets, and that there are only P(R) many sets. So, B(&gamma;) is the set of Borel sets.


Then, you could do something with transfinite induction. Specifically, if you prove:

(1) If your statement holds for B(0)
(2) If your statement holds for anything in C(α) then it holds for anything in B(α)

Then, by transfinite induction, it holds for all B(α).
 
  • #6
jimmysnyder said:
Is the set of inverse images of sets under f a [itex]\sigma[/itex]-algebra?

Bad question. Better would be:

Is the set of sets whose inverse image under f is measurable a [itex]\sigma[/itex]-algebra?
 
  • #7
I think the answer is yes because the inverse image of a union of sets is the union of the inverse images and similarly for intersections. Since the set of measurable sets in the domain is a [itex]\sigma[/itex]-algebra, so is the set of sets in the range whose inverse images are measureable. Hence given (1), the set of sets whose inverse images are measureable is a [itex]\sigma[/itex]-algebra that contains the open sets.
 
  • #8
i.e. one of the most useful remarks i recall from measure theory was " the process of taking inverse images is a boolean sigma homomorphism". ... i.e. it preserves all those operations.
 
  • #9
Thanks. Seems like transfinite induction is very useful. I don't really have any background in ordinal, cardinal, transfinite induction, etc. can anyone suggest where I can read more (preferably free online lecture note) about them?

Before I have a chance to learn them, does anyone know of any proof that avoid using transfinite induction? Thanks.
 
  • #10
You don't actually have to induct, since the borel sigma algebra is defined to be the smallest sigma algebra containing the open sets (strictly speaking I suppose it ought to be the borel sets, but its easy to see that they generate the same borel algebra). So the two definitions are equivalent by fiat.
 

Related to Definitions of Measurable Functions

1. What is a measurable function?

A measurable function is a mathematical function that assigns a numerical value to every point in a given set, called the domain. It is commonly used in statistics and real analysis to describe the behavior of a system or phenomenon.

2. What is the importance of measurable functions in scientific research?

Measurable functions are essential in scientific research as they provide a way to quantify and measure various aspects of a system or phenomenon. They allow for precise and accurate analysis and interpretation of data, making them crucial in many fields of science.

3. Can you give an example of a measurable function?

One example of a measurable function is the temperature of a room over time. The temperature, which is the output or dependent variable, can be measured at different points in time, which represents the input or independent variable. The function maps the time to the temperature, making it measurable.

4. How are measurable functions different from other types of functions?

Measurable functions differ from other types of functions in that they have specific properties that make them suitable for measuring and analyzing data. These properties include being defined over a measurable space, having a measurable domain, and satisfying certain continuity and integrability conditions.

5. What are some applications of measurable functions?

Measurable functions have numerous applications in various fields of science, including physics, economics, and engineering. They are used in statistical models, probability theory, and signal processing, among others, to analyze data and make predictions about the behavior of a system or phenomenon.

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