Proving Equality of Characteristic Functions using Lebesgue Measure?

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Therefore, if µ(EΔF) = 0, the sets have the same measure and the characteristic functions must also be equal almost everywhere. And vice versa, if the characteristic functions are equal almost everywhere, then the set where they differ must have measure 0, meaning µ(EΔF) = 0. So, χ_E = χ_F almost everywhere if and only if µ(EΔF) = 0. Therefore, in summary, for sets E,F \in L, χ_E = χ_F almost everywhere if and only if µ(EΔF) = 0.
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Homework Statement


For sets E,F [tex]\in[/tex] L, show that χ_E = χ_F almost everywhere if and only if µ(EΔF) = 0


where χ_E is the characteristic function w.r.t. E
and µ(EΔF) is the lebesgue measure of the symmetric difference of E and F
and L is the set of lebesgue measurable sets


Homework Equations



A property about real numbers holds almost everywhere if the set of x where it fails to be true has Lebesgue measure 0.


The Attempt at a Solution



I'm really stuck on this. I'm not asking anyone to do it for me, but if anyone could please give me a point in the right direction, that would be great thanks!
 
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  • #2
Isn't the set where the characteristic function of E and F differ equal to the symmetric difference of E and F?
 

1. What is the Lebesgue integral?

The Lebesgue integral is a mathematical concept that extends the traditional Riemann integral to a wider class of functions. It was developed by French mathematician Henri Lebesgue in the early 20th century and is used to calculate the area under a curve or the volume under a surface.

2. How is the Lebesgue integral different from the Riemann integral?

The Lebesgue integral is based on the concept of measure, which is more general than the concept of length used in the Riemann integral. This allows for a wider class of functions to be integrated, including functions that are not continuous.

3. What are the advantages of using the Lebesgue integral?

The Lebesgue integral has several advantages over the Riemann integral. It can handle a wider class of functions, it is more intuitive and easier to understand, and it has better convergence properties. It also allows for the integration of functions with infinite values or infinite domains.

4. What are some applications of the Lebesgue integral?

The Lebesgue integral is used in a variety of fields, including physics, engineering, economics, and probability theory. It is also used in signal processing and image analysis to calculate the area under curves and surfaces, and in machine learning to calculate probabilities and expected values.

5. How is the Lebesgue integral computed?

The Lebesgue integral is computed by dividing the domain of the function into smaller intervals and approximating the area under the curve in each interval. The sum of these approximations gives the total area under the curve. This process is similar to the Riemann sum used in the traditional Riemann integral, but with a more general definition of the area of each interval.

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