Absolutely Continuous Functions Review

Absolutely Continuous Functions Review

We will now review some of the recent material regarding absolutely continuous functions.

(1)
\begin{align} \quad V(P, f) = \sum_{i=1}^{n} | f(x_i) - f(x_{i-1}) | \end{align}
• We said that $f$ is of Bounded Variation on $[a, b]$ if there exists an $M \in \mathbb{R}$, $M > 0$ such that for all partitions $P$ on $[a, b]$ we have that $V(P, f) \leq M$.
• We noted a very important result - every function of bounded variation can be expressed as a difference of two increasing functions.
• On the Absolute Continuity page we said that a function $f : [a, b] \to \mathbb{R}$ is Absolutely Continuous on $[a, b]$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $\{ (x_1, y_1), (x_2,y_2), ..., (x_n, y_n) \}$ is a finite collection of mutually disjoint open subintervals of $[a, b]$ such that $\displaystyle{\sum_{i=1}^{n} (y_i - x_i) < \delta}$ then:
(2)
\begin{align} \quad \sum_{i=1}^{n} | f(y_i) - f(x_i) | < \epsilon \end{align}
• We noted that every absolutely continuous function on $[a, b]$ is uniformly continuous on $[a, b]$ and hence continuous on $[a, b]$. Therefore absolute continuity is a stronger version of continuity.
• On the Functions of Lebesgue Integrals page we began to look at functions of Lebesgue integrals. In particular, if $f$ is Lebesgue integrable on $[a, b]$ we defined the new function $F : [a, b] \to \mathbb{R}$ by:
(3)
\begin{align} \quad F(x) = \int_a^x f(t) \: dt \end{align}
Property 1 If $f$ is Lebesgue integrable on $[a, b]$ then $F$ is uniformly continuous on $[a, b]$. If $f$ is Lebesgue integrable on $[a, b]$ then $F$ is of bounded variation on $[a, b]$.
Lemma 1 If $f$ is a nonnegative Lebesgue integrable function on $E$ and $\displaystyle{\int_E f = 0}$ then $f(x) = 0$ almost everywhere on $[a, b]$. If $f$ is a Lebesgue measurable function and $F(x) = \int_a^x f(t) \: dt$ is such that $F(x) = 0$ for all $x \in [a, b]$ then $f(x) = 0$ almost everywhere on $[a, b]$.
Theorem 1 If $f$ is a bounded Lebesgue integrable function on $[a, b]$ and $\displaystyle{F(x) = \int_a^x f(t) \: dt}$ then $F'(x) = f(x)$ almost everywhere on $[a, b]$. If $f$ is a Lebesgue integrable function on $[a, b]$ and $\displaystyle{F(x) = \int_a^x f(t) \: dt}$ then $F'(x) = f(x)$ almost everywhere on $[a, b]$.
• We used this to prove our major result which classified absolutely continuous functions. We proved that a real-valued function $F$ is absolutely continuous on $[a, b]$ if and only if there exists a Lebesgue integrable function $f$ on $[a, b]$ such that:
(4)
\begin{align} \quad F(x) = \int_a^x f(t) \: dt + F(a) \end{align}