alternative definition of Lebesgue integral, an
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.
- 1 Lieb, E. H., Loss, M., Analysis, AMS, 2001.
|Title||alternative definition of Lebesgue integral, an|
|Date of creation||2013-03-22 17:32:46|
|Last modified on||2013-03-22 17:32:46|
|Last modified by||Gorkem (3644)|