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 $(X,\mathcal{M},\mu )$ be a measure space^{}. Let $f:x\to {\mathbb{R}}^{+}\cup \{0\}$ be a nonnegative measurable function. We will define $\int f\mathit{d}\mu $ in $[0,\mathrm{\infty}]$ and will call it as the Lebesgue integral of $f$.
If there exists a $t>0$ such that $\mu \left(\{x:f(x)>t\}\right)=\mathrm{\infty}$, then we define $\int f\mathit{d}\mu =\mathrm{\infty}.$
Otherwise, assume $$ for all $t\in (0,\mathrm{\infty})$ and let ${F}_{f}(t)=\mu \left(\{x:f(x)>t\}\right)$. ${F}_{f}(t)$ is a monotonically non-increasing function on $(0,\mathrm{\infty})$, therefore its Riemann integral is well defined on any interval $[a,b]\subset (0,\mathrm{\infty})$, so it exists as an improper Riemann integral on $(0,\mathrm{\infty})$. We define
$$\int f\mathit{d}\mu ={\int}_{0}^{\mathrm{\infty}}{F}_{f}(t)\mathit{d}t.$$ |
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 |
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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 |