Fundamental theorem of calculus in terms of Lebesgue integral

In summary, the necessary and sufficient conditions for the equality F(x) = \frac{d}{dx} \int_a^x F(y)dy to hold when the integral is the Lebesgue integral are that the indefinite integral F(x) is absolutely continuous and the derivative F' is Lebesgue integrable. This can be proven using the theorems (*) and (**), with the bulk of the proof relying on the lemma that a nondecreasing, absolutely continuous function with a derivative of 0 almost everywhere is constant. Further clarification on definitions and concepts can be found in more modern texts or through online resources.
  • #1
AxiomOfChoice
533
1
What restrictions must we place on a real-valued function [tex]F[/tex] for

[tex]
F(x) = \frac{d}{dx} \int_a^x F'(y)dy
[/tex]

to hold, where "[itex]\int[/itex]" is the Lebesgue integral?
 
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  • #2
Your question is confusing (also misstated). You have F = F'.
 
  • #3
mathman said:
Your question is confusing (also misstated). You have F = F'.

You're right. Sorry. What I meant to ask was what are necessary and sufficient conditions for the equality

[tex]
F(x) = \frac{d}{dx} \int_a^x F(y)dy
[/tex]

to hold, when the integral is the Lebesgue (not the Riemann) integral.
 
  • #4
I learned the relevant theorems via Kolmogorov & Fomin's Introductory Real Analysis, which is actually a translated and edited version by Richard Silverman. Apparently Silverman added the chapter on differentiation theory, since the material seems to be taken almost directly from Riesz-Nagy's functional analysis text, I think.

Anyways, the first relevant theorem is the following:

(*)The indefinite integral [itex]F(x) = \int_{a}^{x}f(t)\,dt[/itex] of a Lebesgue integrable function f is absolutely continuous.

This direction is pretty easy to prove, and the relevant idea is the continuity of the Lebesgue integral.

The other relevant theorem, which is apparently due to Lebesgue, is the following:

(**)If F is absolutely continuous on [a,b], then the derivative F' is Lebesgue integrable on [a,b], and

[tex]F(x) = F(a) + \int_{a}^{x}F'(t)\,dt.[/tex]

At least the way I learned it, the bulk of this proof rests in the following lemma:

If f is an absolutely continuous nondecreasing function on [a,b] such that f'(x) = 0 almost everywhere, then f is constant.

Combining (*) and (**) gives you a possible characterization of what you're looking for. If you're unclear about the relevant ideas involved, wikipedia is probably your best bet. Feel free to ask me about any particular definitions that might need clarification, since I know Kolmogorov & Fomin sometimes use definitions that are different but equivalent to those found in more modern texts.
 

1. What is the fundamental theorem of calculus in terms of Lebesgue integral?

The fundamental theorem of calculus in terms of Lebesgue integral is a mathematical theorem that relates differentiation and integration. It states that if a function is integrable on a closed interval, then the definite integral of its derivative over that interval is equal to the difference between the function's values at the endpoints of the interval.

2. How is the fundamental theorem of calculus in terms of Lebesgue integral different from the classical fundamental theorem of calculus?

The classical fundamental theorem of calculus only applies to continuous functions, while the Lebesgue integral version applies to a broader class of functions, including those that are not continuous. Additionally, the Lebesgue integral version uses a different definition of the integral, which is based on a different measure called the Lebesgue measure.

3. Why is the fundamental theorem of calculus in terms of Lebesgue integral important in mathematics?

This theorem is a fundamental result in analysis and is used in many areas of mathematics, such as calculus, differential equations, and probability theory. It allows for the evaluation of integrals of a wide range of functions and is essential for understanding the relationship between differentiation and integration.

4. Can the fundamental theorem of calculus in terms of Lebesgue integral be extended to higher dimensions?

Yes, the fundamental theorem of calculus can be extended to higher dimensions using the concept of multivariable integration. In this case, the theorem states that the integral of a partial derivative of a multivariable function over a region is equal to the difference between the function's values at the boundary of the region.

5. Are there any practical applications of the fundamental theorem of calculus in terms of Lebesgue integral?

Yes, the fundamental theorem of calculus in terms of Lebesgue integral has many practical applications in physics, engineering, and economics. It is used to solve problems involving rates of change, such as finding the velocity of an object or the rate of change of a quantity over time. It is also used in optimization problems and in the calculation of areas and volumes.

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