Question about baire class 1 functions

Your second suggestion is interesting, as it uses the Borel hierarchy to define the Baire classes. However, it may be more difficult to use practically.In summary, the conversation discusses a question related to measure theory and the Baire and Borel classes of functions. The solution involves constructing a sequence of continuous functions that converges to the given function, or using the fact that the function belongs to a Baire class if its inverse set belongs to a Borel class. The discussion also notes that the Baire and Borel classes are not necessarily disjoint.
  • #1
hermanni
25
0
Hi all,
I'm a graduate student and while I was reading about measure theory I stuck at this question , can anyone help?

Let the function f: R->R be continuous except on a countable set. Show that f belongs to
Baire class 1 of functions.

For the solution I see 2 ways :
1. We need to construct a sequence f{n} from Baire -0 (set of continuous functions)
that converges to f.
2. Or we can use the fact that f belongs to Baire-n iff f^-1 (O) belongs to
Borel-n+1 the set for every O open.

Regards, h.
 
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  • #2
First of all, your classes are not disjoint. This is important to know, as some others exclude Baire 0 classes from Baire 1 classe, which is not the case here.

Then your first suggestion should immediately lead to the result, as you only need pointwise convergence.
 

FAQ: Question about baire class 1 functions

What is a Baire class 1 function?

A Baire class 1 function is a type of real-valued function that is continuous almost everywhere, meaning it is continuous on a dense set of points. This class of functions is important in functional analysis and is named after French mathematician René-Louis Baire.

How is a Baire class 1 function different from other types of functions?

A Baire class 1 function is different from other types of functions, such as continuous or differentiable functions, because it is only required to be continuous on a dense set of points rather than on the entire domain. This means that it may not be continuous at every point, but it is still considered a well-behaved function.

3. What is the significance of Baire class 1 functions in mathematics?

Baire class 1 functions are significant in mathematics because they are a fundamental class of functions in functional analysis, a branch of mathematics that studies vector spaces, operators, and infinite-dimensional spaces. They also have applications in other areas such as topology and measure theory.

4. Can Baire class 1 functions be extended to higher dimensions?

Yes, Baire class 1 functions can be extended to higher dimensions. In multi-dimensional spaces, a function is considered to be in Baire class 1 if it is continuous almost everywhere, meaning it is continuous on a dense set of points in each coordinate direction. This is known as Baire class 1 in the sense of coordinate functions.

5. Are there any other classes of functions similar to Baire class 1?

Yes, there are other classes of functions that are similar to Baire class 1. These include Baire class 2, Baire class 3, and so on, which require the function to be continuous almost everywhere in a stronger sense. There are also related classes of functions such as Borel measurable functions and Lebesgue measurable functions that have similar properties.

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