definition of Lebesgue integral of measurable extended real function over measurable subset of measure space
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of measurable map between measurable spaces.
- The reader knows a definition of extended Euclidean topological space.
- The reader knows a definition of Borel \(\sigma\)-algebra of topological space.
- The reader knows a definition of simple map.
Target Context
- The reader will have a definition of Lebesgue integral of measurable extended real function over measurable subset of measure space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( (M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\( \overline{\mathbb{R}}\): \(= \text{ the extended Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\( \mathbb{R}\): \(= \text{ the Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\( f\): \(: M \to \overline{\mathbb{R}}\), \(\in \{\text{ the measurable maps }\}\)
\( a\): \(\in A\)
\( \chi_a\): \(= \text{ the characteristic function of } a\)
\( f^+\): \(: M \to [0, \infty], s \mapsto max (0, f (s))\)
\( f^-\): \(: M \to [0, \infty], s \mapsto - min (0, f (s))\)
\( P^+\): \(= \{f \in \{\text{ the measurable functions from } M \text{ into } \mathbb{R}\} \cap \{\text{ the simple maps }\} \vert 0 \le f\}\)
\(*\int_a f d \mu\): \(= \int_M \chi_a f^+ d \mu - \int_M \chi_a f^- d \mu\)
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Conditions:
for \(g := f^+ \text{ or } f^-\), \(\int_M \chi_a g d \mu = sup_{h \in P^+ \text{ such that } h \le \chi_a g} \int_M h d \mu\), where \(\int_M h d \mu\) means for \(h (M) = \{r_1, ..., r_n\}\), \(\sum_{j \in \{1, ..., n\}} r_j \mu (h^{-1} (r_j))\)
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When \(\int_M \chi_a f^+ d \mu = \infty\) and \(\int_M \chi_a f^- d \mu = \infty\), it is not well-define, but otherwise, it is well-defined.
2: Note
Let us see that \(\int_M \chi_a g d \mu\) is well-defined.
\(\{h \in P^+ \text{ such that } h \le \chi_a g\}\) is not empty, because at least, \(h = 0\) is in it.
So, \(sup_{h \in P^+ \text{ such that } h \le \chi_a g} \int_M h d \mu\) exists as a nature of the real numbers set (may be \(\infty\)).
But why does \(f\) need to be measurable?, which arises as a question because the definition does not need \(f\) to be measurable: what are need to be measurable are only \(h\) s.
In fact, it is possible to define integral for any non-measurable function.
But that integral would not be guaranteed to satisfy some natural properties like \(\int_a (f_1 + f_2) d \mu = \int_a f_1 d \mu + \int_a f_2 d \mu\), because that uses the fact that for any measurable \(f\) into \([0, \infty]\), there is a increasing sequence of some simple measurable functions into \([0, \infty)\), \(f_1, f_2, ...\), such that \(f_1 \le f_2 \le ...\) and \(lim_j f_j (x) = f (x)\), which is not guaranteed to hold for a non-measurable \(f\).
So, usually, \(f\) is required to be measurable.
\(a\) is required to belong to \(A\) for requiring \(\chi_a f\) to be measurable: (refer to the proposition that for any map into any Euclidean set and any characteristic function, the preimage of any subset under the map multiplied by the characteristic function is this).
