How Do You Calculate the Lebesgue Integral of e^x from 1 to 10?

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In summary, the Lebesgue integral is equivalent to the Riemann integral on an interval if the function is Riemann integrable on that interval. Additionally, on the interval X=(0,t), the derivative of the Lebesgue integral is equal to the function f(t) except possibly on a set of measure 0. This is based on certain hypotheses and can be seen on pages 11 and 47-48 of Riesz Nagy.
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eljose
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let be the Lebesgue integral with a meassure [tex]\mu[/tex] then if we call this integral..

[tex]\int_{X}fd\mu=I [/tex]

my questions are..how would you calculate the integral of f(x)=exp(x) from 1 to 10?..another question let be the lebesgue integral on the interval X=(0,t) would be true that:

[tex](\frac{d}{dt}\int_{X}fd\mu=f(t) [/tex]?

thanks...
 
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eljose said:
let be the Lebesgue integral with a meassure [tex]\mu[/tex] then if we call this integral..
[tex]\int_{X}fd\mu=I [/tex]
my questions are..how would you calculate the integral of f(x)=exp(x) from 1 to 10?
The same way you would a Riemann integral. If a function is Riemann integrable on an interval, then it is Lebesque integrable on that interval and the two integrals are the same.

..another question let be the lebesgue integral on the interval X=(0,t) would be true that:
[tex](\frac{d}{dt}\int_{X}fd\mu=f(t) [/tex]?
thanks...
Except possibly on a set of measure 0, yes.
 
  • #3
you need some hypotheses of course, but if f is "summable" then it is the derivative a.e. of its indefinite integral. see pages 11 and 47, 48 of Riesz Nagy.
 

What is the Lebesgue integral?

The Lebesgue integral is a mathematical concept used to calculate the area under a curve or the volume under a surface. It was developed by French mathematician Henri Lebesgue in the early 20th century as a more general and powerful alternative to the Riemann integral. It is widely used in various fields of mathematics, including analysis, probability, and measure theory.

How is the Lebesgue integral different from the Riemann integral?

While both the Lebesgue and Riemann integrals are used to calculate the area under a curve, they differ in their approach. The Riemann integral divides the area into small rectangles and sums their areas, while the Lebesgue integral divides the area into smaller subsets and sums their contributions. This allows the Lebesgue integral to handle a wider range of functions, including those that are not continuous or have infinite discontinuities.

What are the main properties of the Lebesgue integral?

The Lebesgue integral has several important properties that make it a valuable tool in mathematical analysis. These include linearity, monotonicity, and additivity. It also allows for the interchange of the order of integration and has a powerful convergence theorem, the Lebesgue dominated convergence theorem, which allows for the calculation of the integral of a limit of functions.

What are some applications of the Lebesgue integral?

The Lebesgue integral has many practical applications, particularly in probability theory and statistics. It is used to calculate probabilities of events, expected values, and moments of random variables. It is also used in physics and engineering to calculate various physical quantities, such as energy, mass, and volume. In addition, the Lebesgue integral is used in signal processing, image analysis, and other areas of applied mathematics.

What are the challenges in understanding and using the Lebesgue integral?

The Lebesgue integral can be challenging to understand and use, as it requires a solid foundation in mathematical analysis and measure theory. It also involves abstract concepts, such as measure, measurable sets, and integrable functions. Additionally, calculating the Lebesgue integral can be computationally intensive, especially for higher-dimensional problems. Proper understanding and application of the Lebesgue integral require practice and a good understanding of its properties and techniques.

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