alternative definition of Lebesgue integral, an
The standard way of defining Lebesgue integral is first to define it for simple functions
, and then to take limits for arbitrary positive measurable functions
.
There is also another way which uses the Riemann integral [1].
Let be a measure space. Let be a nonnegative measurable function. We will define in and will call it as the Lebesgue integral of .
If there exists a such that , then we define
Otherwise, assume for all and let . is a monotonically non-increasing function on , therefore its Riemann integral is well defined on any interval , so it exists as an improper Riemann integral on . We define
The definition can be extended first to real-valued functions, then complex valued functions as usual.
References
-
1
Lieb, E. H., Loss, M., Analysis
, AMS, 2001.
Title | alternative definition of Lebesgue integral, an |
---|---|
Canonical name | AlternativeDefinitionOfLebesgueIntegralAn |
Date of creation | 2013-03-22 17:32:46 |
Last modified on | 2013-03-22 17:32:46 |
Owner | Gorkem (3644) |
Last modified by | Gorkem (3644) |
Numerical id | 6 |
Author | Gorkem (3644) |
Entry type | Definition |
Classification | msc 26A42 |
Classification | msc 28A25 |