Question about images of measurable functions

In summary, the statement is trying to prove that the set B, defined as the set of real numbers y for which the inverse image of y under the measurable function f has a measure greater than 0, is countable. The conversation suggests using the fact that any uncountable subset of a complete metric space has a condensation point, but the speaker is unsure of how to derive a contradiction from this point. The other person suggests intersecting the sets with a finite measure interval, such as [0,1], to show that the sum of the measures of all the sets in B intersected with [0,1] is less than or equal to 1, making it impossible for an uncountable number of measures to be
  • #1
bbkrsen585
11
0
I want to prove the following.

Statement: Given that f is measurable, let
B = {y [tex]\in [/tex] ℝ : μ{f^(-1)(y)} > 0}. I want to prove that B is a countable set.

(to clarify the f^(-1)(y) is the inverse image of y; also μ stands for measure)

Please set me in the right direction. I would greatly appreciate it. Thanks!
 
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  • #2
Use the fact that any uncountable subset of a complete metric space has a condensation point (a point whose every neighborhood contains uncountably many points of the set).
 
  • #3
So I've thought about that, but here's the thing. Suppose I assume that B is uncountable. Then, for each y in B, there is a corresponding open image that has measure greater than zero. Since we have uncountably many y's (by assumption), there would be an uncountably many number of open images with measure greater than zero, which implies that the domain is of infinite measure. I'm not sure where to derive the contradiction from this point. Thanks.
 
  • #4
bbkrsen585 said:
So I've thought about that, but here's the thing. Suppose I assume that B is uncountable. Then, for each y in B, there is a corresponding open image that has measure greater than zero. Since we have uncountably many y's (by assumption), there would be an uncountably many number of open images with measure greater than zero, which implies that the domain is of infinite measure. I'm not sure where to derive the contradiction from this point. Thanks.

Intersect your sets with, say, [0,1] which has finite measure. The sum of the measures of all of the sets in B intersected with [0,1] is less than or equal to 1, right? Can an uncountable number of the measures be positive?
 

Related to Question about images of measurable functions

1. What is a measurable function?

A measurable function is a function in mathematics that maps elements from a measurable space to a measurable space. In simpler terms, it is a function that takes in inputs and produces outputs that can be measured or assigned a numerical value.

2. How do you determine if a function is measurable?

In order for a function to be measurable, the preimage of any measurable set in the output space must be a measurable set in the input space. This means that the inverse image of any interval must be a measurable set.

3. What is the importance of measurable functions in mathematics?

Measurable functions are important in various branches of mathematics, particularly in measure theory and probability. They are used to describe and model real-world phenomena, and are essential in understanding and solving complex mathematical problems.

4. Can all functions be measurable?

No, not all functions are measurable. There are some functions that are non-measurable, meaning they do not follow the criteria for being measurable, such as the Dirichlet function.

5. What are some real-life examples of measurable functions?

Measurable functions can be found in many real-world applications, such as in economics where they are used to model consumer demand and utility, in physics to describe physical processes and phenomena, and in engineering to analyze and design systems. They are also used in data analysis and statistics to analyze and interpret data.

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