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.