Daniell Integral: Overview & Explanation

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In summary, The Daniell integral is a method for defining integrals for functions on a 3-D volume without needing measure theory by defining a set of elementary functions and a continuous linear functional on those functions and then extending it to a larger class of functions. This is similar to how real numbers are extended to irrational numbers by taking limits of rational sequences.
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Karlisbad
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Could someone explain some info about this kind of integral??..i watched at wikipedia that this allowed you to find a measure for spaces without recurring to Measure Theory...i looked at Wikipedia but found no further info..:frown: :frown: cold someone provide a valuable web-link or similar or explain (in easy concepts) what's all this about?, thanks:tongue2:
 
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I thought the wikipedia article did a pretty good job explaining it. Do you have any specific questions? Otherwise, all I can suggest is searching google.
 
  • #3
The question is..let be the next integral:

[tex] \int_{V}d\mu f(X) [/tex] V is a 3-D volume and X=(x,y,z) of course we have the problem in defining the meassure [tex] \mu [/tex] but i think Daniell integral can avoid this problem but how??...
 
  • #4
As discussed in the article, the idea is to define a set of elementary functions, a continuous linear functional on those functions, and then extend by continuity to a larger class of functions.

In this case, the class of elementary functions can be taken as the set of continuous functions on some compact subset of Rn, and the linear functional is the ordinary Riemann integral. This is continuous, in the sense that if a sequence of non-negative continuous functions converges pointwise to zero, then their integrals converge to zero. Thus we can uniquely define an integral for any function which is the pointwise limit of a sequence of continuous functions by taking the limit of their integrals, and apparently this recovers the Lebesgue integral. Alternatively, starting with the step functions (linear combinations of the characteristic functions of intervals) and taking their integral as the area underneath them also gives back the Lebesgue integral.

You can think of this as being analgous to defining [itex]a^r[/itex] for real numbers r by first defining it for rational r by [itex]a^{p/q}=\sqrt[q]{a^p}[/itex], and then extending to all r by continuity (ie, taking the limit of the function evaluated on rational sequences approaching r).
 
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What is the Daniell Integral?

The Daniell Integral is a mathematical concept used in calculus to find the area under a curve. It was developed by the mathematician Percy John Daniell in the early 20th century.

How does the Daniell Integral differ from other types of integrals?

The Daniell Integral differs from other types of integrals, such as the Riemann Integral, in that it allows for integration of functions that are not continuous. It also has a more general definition, making it applicable to a wider range of functions.

What is the process for calculating the Daniell Integral?

To calculate the Daniell Integral, you must first partition the interval of integration into smaller subintervals. Then, you must take the upper and lower sums of the function on each subinterval. Finally, you take the limit as the subintervals become infinitely small to find the Daniell Integral.

What are some real-world applications of the Daniell Integral?

The Daniell Integral has many applications in physics and engineering, such as calculating work done by a varying force, calculating the center of mass of a non-uniform object, and determining the total charge on a continuous distribution of charge.

Are there any limitations to using the Daniell Integral?

One limitation of the Daniell Integral is that it can only be applied to functions that are bounded on the interval of integration. It also requires a more complex calculation process compared to other types of integrals, which may make it less practical in certain situations.

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