User:Caliburn/s/mt/Riemann-Lebesgue Theorem/Proof 2

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Proof

From Condition for Darboux Integrability, we have:

for all $\epsilon > 0$ there exists a finite subdivision $P$ such that $\map U {f, P} - \map L {f, P} < \epsilon$

where:

$\map U {f, P}$ is the upper Darboux sum of $f$ with respect to $P$
$\map L {f, P}$ is the lower Darboux sum of $f$ with respect to $P$.


We now construct a sequence of finite subdivisions $\sequence {P_n}_{n \mathop \in \N}$.

Let $P_1$ be such that:

$\map U {f, P_1} - \map L {f, P_1} < 1$

For $n > 1$, let $Q_n$ be such that:

$\ds \map U {f, Q_n} - \map L {f, Q_n} < \frac 1 n$

and:

$P_n = Q_n \cup P_{n - 1}$

Since $P_n$ is a refinement of $Q_n$, we have:

$\map U {f, P_n} \le \map U {f, Q_n}$

and:

$\map L {f, P_n} \ge \map L {f, Q_n}$

from Upper Sum of Refinement and Lower Sum of Refinement.

We also have that:

$P_{n - 1} \subseteq P_n$

for $n > 1$, so the sequence $\sequence {P_n}_{n \mathop \in \N}$ is increasing.

For each $n$, write:

$P_n = \set {a_1, a_2, \ldots, a_{k_n} }$

with:

$a = a_1 < a_2 < \ldots < a_{k_n} = b$


For each $1 \le i \le k_n - 1$ define:

$m_i = \inf \set {\map f x : x \in \hointl {a_i} {a_{i + 1} } }$

and:

$M_i = \set {\map f x : x \in \hointl {a_i} {a_{i + 1} } }$

From Continuous Function on Compact Subspace of Euclidean Space is Bounded, these quantities are finite for each $i$.

For each $n$, define the function $g_n : \closedint a b \to \R$ by:

$\ds g_n = \sum_{i \mathop = 1}^{k_n - 1} m_i \chi_{\hointl {a_i} {a_{i + 1} } }$

and the function $h_n : \closedint a b \to \R$ by:

$\ds h_n = \sum_{i \mathop = 1}^{k_n - 1} M_i \chi_{\hointl {a_i} {a_{i + 1} } }$

Note that for each $n$ we have:

$g_n$ is simple

and:

$h_n$ is simple.

So from Simple Function is Measurable, we have:

$g_n$ is $\lambda$-measurable

and:

$h_n$ is $\lambda$-measurable.

From Integral of Characteristic Function: Corollary, we have:

$\ds \int \chi_{\hointl {a_i} {a_{i + 1} } } \rd \lambda = \map \lambda {\hointl {a_i} {a_{i + 1} } } = a_{i + 1} - a_i$

So, from Integral of Integrable Function is Homogeneous, we have:

$\ds \int M_i \chi_{\hointl {a_i} {a_{i + 1} } } \rd \lambda = M_i \paren {a_{i + 1} - a_i}$

and:

$\ds \int m_i \chi_{\hointl {a_i} {a_{i + 1} } } \rd \lambda = m_i \paren {a_{i + 1} - a_i}$

So, we have:

\(\ds \int g_n \rd \lambda\) \(=\) \(\ds \int \paren {\sum_{i \mathop = 1}^{k_n - 1} m_i \chi_{\hointl {a_i} {a_{i + 1} } } } \rd \lambda\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{k_n - 1} \paren {\int m_i \chi_{\hointl {a_i} {a_{i + 1} } } \rd \lambda}\) Integral of Integrable Function is Additive
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{k_n - 1} m_i \paren {a_{i + 1} - a_i}\)
\(\ds \) \(=\) \(\ds \map L {f, P_n}\) Definition of Lower Darboux Sum

and:

\(\ds \int h_n \rd \lambda\) \(=\) \(\ds \int \paren {\sum_{i \mathop = 1}^{k_n - 1} M_i \chi_{\hointl {a_i} {a_{i + 1} } } } \rd \lambda\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{k_n - 1} \paren {\int M_i \chi_{\hointl {a_i} {a_{i + 1} } } \rd \lambda}\) Integral of Integrable Function is Additive
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{k_n - 1} M_i \paren {a_{i + 1} - a_i}\)
\(\ds \) \(=\) \(\ds \map U {f, P_n}\) Definition of Upper Darboux Sum
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