Lebesgue Integration: Finite Measure Not Sufficient

In summary, Lebesgue Integration is a mathematical theory and technique that allows for the calculation of area and volume under curves and surfaces. It is named after French mathematician Henri Lebesgue and is a more advanced form of integration compared to Riemann Integration. Finite measure is not sufficient in Lebesgue Integration as it is limited to certain types of functions, while Lebesgue Integration allows for a wider range. It differs from Riemann Integration in its approach and the types of functions it can integrate. Lebesgue Integration has various applications in mathematics, physics, engineering, and other fields, such as probability and measure theory, solving differential equations, and signal processing. It is defined using the Lebesgue measure, which provides a
  • #1
wayneckm
68
0
Hello all,

Here is my question:

Suppose a measureable space [tex] (S,\mathcal{S},\mu) [/tex] with [tex] \mu(S) < \infty [/tex] and [tex] f : S \mapsto [0,\infty) [/tex], this is not yet sufficient to ensure [tex] \int_{S} f d \mu < \infty [/tex].

Am I correct?
 
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  • #2
S=(0,1) and f=1/x
 
  • #3
Yes. Consider the measure defined on R by

[tex]\mu(E) =\int_E \frac 1 {1+x^2}\ dx[/tex]

for Lebesgue measurable E. Let f(x) = 1/x2. Then

[tex]\int_R \frac 1 {x^2}\cdot \frac 1 {1+x^2}\ dx \ge \int_{-1}^1 \frac 1 {x^2}
\cdot \frac 1 2\ dx =\infty[/tex]
 

FAQ: Lebesgue Integration: Finite Measure Not Sufficient

What is Lebesgue Integration?

Lebesgue Integration is a mathematical theory and technique used to calculate the area under a curve or the volume under a surface. It is named after French mathematician Henri Lebesgue and is a more advanced form of integration compared to the traditional Riemann Integration.

Why is finite measure not sufficient in Lebesgue Integration?

Finite measure is not sufficient in Lebesgue Integration because it is limited to only certain types of functions and does not consider more general cases. Lebesgue Integration allows for a wider range of functions and provides a more flexible and powerful framework for integration.

What is the difference between Riemann Integration and Lebesgue Integration?

Riemann Integration is based on approximating the area under a curve using rectangles, while Lebesgue Integration is based on approximating the area using smaller and smaller intervals. Lebesgue Integration also allows for the integration of a wider range of functions compared to Riemann Integration.

What are some applications of Lebesgue Integration?

Lebesgue Integration has various applications in mathematics, physics, engineering, and other fields. It is used to calculate probabilities, areas and volumes in probability and measure theory, to solve differential equations in physics and engineering, and in various other areas such as signal processing, image processing, and statistics.

How is Lebesgue Integration related to the Lebesgue measure?

The Lebesgue Integration is defined using the Lebesgue measure, which is a measure of the size of a set in a given space. The Lebesgue measure provides a way to assign a measure to a set in a more general and flexible way compared to traditional measures, allowing for the integration of a wider range of functions.

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