Ex: Let \(X=[0,1]\), \(f(x)=r\) for each \(r \in Q \subset X\) and \(f(x)=0\) otherwise. Riemann Integral lower limit is \(0\) and largest value is \(1\). However, using measurable set as \(Q \cap X\), the Lesbegue integral has value and it is zero.
Thus Lesbegue Integral deals with larger classes of functions.
The Leabesgue Integral has the following nice property while Riemann Integral doesn’t have.
Lesbesgue’s Montone Convergence Theorem: Let \(\{f_n\}\) be be a sequence of measurable functions on \(X\) and suppose that Then \(f\) is measurable, and \[\begin{equation*} \int_X f_n d\mu \rightarrow \int_X f d\mu \end{equation*}\] PROOF: Since \(\int f_n \leq \int f_{n+1}\), there exists \(\alpha \in [0,\infty]\) such that \[\begin{equation} \int_X f_n d\mu \rightarrow \alpha \text{ as } n \rightarrow \infty \end{equation}\] Because \(sup(f_n)\) and \(lim sup \{f_n\}\) as \(n \rightarrow \infty\) are both measurable, \(f\) is measurable. Since \(f_n \leq f\), we have \(\int f_n \leq \int f\) for every \(n\). Hence, \[\begin{equation} \alpha \leq \int_E f d\mu \end{equation}\] All that needs to be shown is \(\alpha \geq \int_E f d\mu\). Let \(s\) be any simple measurable functions such that \(0 \leq s \leq f\), let \(c\) be a constant with \(0 < c < 1\), and define \[\begin{equation*} E_n = \{ x:f_n(x) \geq cf(x) \} \text{ } n=1,2,3,\cdots \end{equation*}\] Each \(E_n\) us neasurable, \(E_1 \subset E_2 \subset E_3 \subset \cdots\) and \(X=\cup E_n\). Also \[\begin{equation*} \int_E f_n d\mu \geq \int_{E_n} f_n d\mu \geq c \int_{E_n}s d\mu \text{ } n=1,2,\cdots \end{equation*}\] As \(n \rightarrow \infty\) Then, \[\begin{equation} \alpha \geq c \int_X s d\mu \end{equation}\] Since above equation holds for every \(c < 1\), we have \[\begin{equation*} \alpha \geq \int_X s d\mu \end{equation*}\] for every measurable \(s\) satisfying \(0 \leq s \leq f\), so that \[\begin{equation} \alpha \geq \int_X f d\mu \end{equation}\] Conclusion follows.
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