Lecture 9

Exam next week.

Transition to new book.

Continue on Chapter 6

Integrable Functions

Theorem 6.11

Suppose f\in \mathscr{R}(\alpha) on [a, b], m\leq f(x)\leq M for all x\in [a, b], and \phi is continuous on [m, M], and let h(x)=\phi(f(x)) on [a, b]. Then h\in \mathscr{R}(\alpha) on [a, b].

Proof:

Since \phi is uniformly continuous on [m, M], for any \epsilon > 0, there exists a \delta > 0 such that if s, t\in [m, M] and |s-t| < \delta, then |\phi(s)-\phi(t)| < \epsilon.

Since f\in \mathscr{R}(\alpha) on [a, b], we can find a partition P=\{x_0, x_1, \cdots, x_n\} of [a, b] such that U(f, P, \alpha)-L(f, P, \alpha) < \epsilon \delta.

Set M_i=\sup \{f(x): x\in [x_{i-1}, x_i]\} and m_i=\inf \{f(x): x\in [x_{i-1}, x_i]\}. M_i^*=\sup \{h(x): x\in [x_{i-1}, x_i]\} and m_i^*=\inf \{h(x): x\in [x_{i-1}, x_i]\}.

We call a index i good if M_i-m_i < \delta.

If i is good, then \forall x, y\in [x_{i-1}, x_i], |f(x)-f(y)| < \delta and by uniform continuity of \phi, |\phi(f(x))-\phi(f(y))| < \epsilon.

Therefore, |M_i^*-m_i^*| < \epsilon.

If i is bad, then M_i-m_i\geq \delta.

Notice that


\begin{aligned}
\delta\sum_{i\in\text{bad}} \Delta \alpha_i &\leq \sum_{i\in\text{bad}} (M_i-m_i) \Delta \alpha_i \\
&\leq \sum_{i=1}^n (M_i-m_i) \Delta \alpha_i \\
&\leq U(f, P, \alpha)-L(f, P, \alpha) \\
&< \epsilon\delta
\end{aligned}

Therefore, \sum_{i=1}^n (M_i^*-m_i^*) \Delta \alpha_i < \epsilon.

So,


\begin{aligned}
U(P,h,\alpha)-L(P,h,\alpha) &= \sum_{i=1}^n (M_i^*-m_i^*) \Delta \alpha_i \\
&\leq \sum_{i\in\text{good}} \epsilon \Delta \alpha_i + \sum_{i\in\text{bad}}2 \sup \{|h(x)-h(y)|: x, y\in [x_{i-1}, x_i]\} \Delta \alpha_i \\
&\leq \epsilon [\alpha(b)-\alpha(a)] + 2\epsilon \sup \{|h(x)-h(y)|: x, y\in [a, b]\}\\
\end{aligned}

Since \epsilon is arbitrary, h\in \mathscr{R}(\alpha) on [a, b].

EOP

Properties of Integrable Functions

Theorem 6.12

Let f,g\in \mathscr{R}(\alpha) on [a, b].

(a) f+g\in \mathscr{R}(\alpha) on [a, b], \int_a^b (f+g)d\alpha = \int_a^b f d\alpha + \int_a^b g d\alpha. (Linearity of the integral)

If c\in \mathbb{R}, then cf\in \mathscr{R}(\alpha) on [a, b], and \int_a^b cf d\alpha = c\int_a^b f d\alpha.

(b) If f(x)\leq g(x),\forall x\in [a, b], then \int_a^b f d\alpha \leq \int_a^b g d\alpha.

(d) If |f(x)|\leq M, then |\int_a^b f d\alpha| \leq M(\alpha(b)-\alpha(a)).

(e) If f\in \mathscr{R}(\beta) then f\in \mathscr{R}(\alpha+\beta) and \int_a^b f d(\alpha+\beta) = \int_a^b f d\alpha + \int_a^b f d\beta.

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