What values of p make \frac{1}{x^\alpha + x^\beta} integrable on (0,\infty)?

In summary, Lebesgue integration is a mathematical technique developed by Henri Lebesgue in the early 20th century for finding the area or volume under a curve or surface in a multi-dimensional space. It is a more general approach than Riemann integration, as it can handle a wider range of functions and provides a more intuitive understanding of integration. Unlike Riemann integration, which partitions the domain of a function, Lebesgue integration partitions the range of a function. The Lebesgue measure is used to assign a value to sets in a multi-dimensional space, and Lebesgue integration has many applications in mathematics, physics, engineering, economics, and finance.
  • #1
AxiomOfChoice
533
1
I'm working through some old prelim problems, and one of them has me stumped:

"For [itex]0 < \alpha < \beta < \infty[/itex], for which positive real numbers [itex]p[/itex] do we have

[tex]
\frac{1}{x^\alpha + x^\beta} \in L^p (0,\infty)
[/tex]
 
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  • #2
Analyze separately how it behaves near 0 and near infinity.
 

FAQ: What values of p make \frac{1}{x^\alpha + x^\beta} integrable on (0,\infty)?

What is Lebesgue integration?

Lebesgue integration is a mathematical technique used to find the area under a curve or the volume under a surface in a multi-dimensional space. It was developed by French mathematician Henri Lebesgue in the early 20th century and is a more general approach to integration than the traditional Riemann integration.

What are the advantages of Lebesgue integration?

Lebesgue integration allows for a more flexible approach to integration than Riemann integration. It can handle a wider range of functions, including those that are not continuous. It also provides a more intuitive understanding of integration as a measure of the size of a set.

How is Lebesgue integration different from Riemann integration?

Unlike Riemann integration, which partitions the domain of a function into small intervals, Lebesgue integration partitions the range of the function into small sets. This allows for a more general approach to integration that can handle a wider range of functions.

What is the Lebesgue measure?

The Lebesgue measure is a mathematical concept used in Lebesgue integration to measure the size of sets in a multi-dimensional space. It assigns a value to each set based on its "volume" or "area" in that space.

What are some applications of Lebesgue integration?

Lebesgue integration has many applications in mathematics, physics, and engineering. It is used to solve problems in areas such as probability theory, signal processing, and differential equations, and has also been applied to problems in economics and finance.

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