Integrals of Upper Functions on General Intervals Review

# Integrals of Upper Functions on General Intervals Review

We will now review some of the recent material regarding upper functions and integrals of upper functions.

• Recall from the Upper Functions and Integrals of Upper Functions page that a function $f$ is said to be an Upper Function on the interval $I$ if there exists an increasing sequence of step functions $(f_n(x))_{n=1}^{\infty}$ that converges to $f$ almost everywhere on $I$ and such that $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ is finite.
• If $f$ is an upper function on $I$ and $(f_n(x))_{n=1}^{\infty}$ is such a sequence, then we say that $(f_n(x))_{n=1}^{\infty}$ is a Generating Sequence for $f$ on $I$.
• Furthermore, the Set of All Upper Functions on $I$ is denoted $U(I)$.
• If $f \in U(I)$ then we define the Integral of $f$ on $I$ to be:
(1)
\begin{align} \quad \int_I f(x) \: dx = \lim_{n \to \infty} \int_I f_n(x) \: dx \end{align}
• We noted that this integral is always well defined. If $(f_n(x))_{n=1}^{\infty}$ and $(f_n^*(x))_{n=1}^{\infty}$ are both generating sequences for $f$ then:
(2)
\begin{align} \quad \lim_{n \to \infty} \int_I f_n(x) \: dx = \lim_{n \to \infty} \int_I f_n^*(x) \: dx \end{align}
(3)
\begin{align} \quad \int_I [f(x) + g(x)] \: dx = \int_I f(x) \: dx + \int_I g(x) \: dx \end{align}
• Furthermore, if $c \geq 0$ then we also have that:
(4)
\begin{align} \quad \int_I cf(x) \: dx = c \int_I f(x) \: dx \end{align}
• In general though, if $f \in U(I)$ then $-f$ may not be in $U(I)$.
(5)
\begin{align} \quad \int_I f(x) \: dx \leq \int_I g(x) \: dx \end{align}
• We then looked at a corollary which said that if $f, g \in U(I)$ and $f(x) = g(x)$ almost everywhere on $I$ then:
(6)
\begin{align} \quad \int_I f(x) \: dx = \int_I g(x) \: dx \end{align}
(7)
\begin{align} \quad \int_I f(x) \: dx = \int_{I_1} f(x) \: dx + \int_{I_2} f(x) \: dx \end{align}
• Furthermore if $I = I_1 \cup I_2$ as above, $g \in U(I_1)$, $h \in U(I_2)$, and $\displaystyle{f(x) = \left\{\begin{matrix} g(x) & x \in I_1\\ h(x) & x \in I_2 \setminus \{I_1 \cap I_2\} \end{matrix}\right.}$ then $f \in U(I)$ and furthermore:
(8)
\begin{align} \quad \int_I f(x) \: dx = \int_{I_1} g(x) \: dx + \int_{I_2} h(x) \: dx \end{align}
• Afterwards, on the The Maximum and Minimum Functions of Two Functions we said that if $f$ and $g$ are functions defined on an interval $I$ then the maximum and minimum functions of $f$ and $g$ denoted $\max (f, g)$ and $\min (f, g)$ are defined for all $x \in I$ by:
(9)
\begin{align} \quad \max (f, g)(x) = \max \{ f(x), g(x) \} \quad \mathrm{and} \quad \min (f, g)(x) = \min \{ f(x), g(x) \} \end{align}
• We proved three very important properties. For $f$, $g$, and $h$ as functions defined on $I$ we have that:
(10)
\begin{align} \quad \min (f, g) \leq \max (f, g) \end{align}
(11)
\begin{align} \quad \max (f, g) + \min (f, g) = f + g \end{align}
(12)
\begin{align} \quad \max (f + h, g + h) = \max (f, g) + h \quad \mathrm{and} \quad \min (f + h, g+ h) = \min (f, g) + h \end{align}
• On the Basic Theorems Regarding the Maximum and Minimum Functions of Two Functions page we looked at two more results regarding the maximum and minimum functions defined above. We saw that if $(f_n(x))_{n=1}^{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ were increasing sequences of functions then $(\max (f_n, g_n))_{n=1}^{\infty}$ and $(\min (f_n, g_n))_{n=1}^{\infty}$ were also increasing sequences of functions.
• Furthermore, if $(f_n(x))_{n=1}^{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ are both increasing sequences of function that converge to $f$ and $g$ almost everywhere on $I$ then $(\max (f_n, g_n))_{n=1}^{\infty}$ and $(\min (f_n, g_n))_{n=1}^{\infty}$ converge (increasingly) to $\max (f, g)$ and $\min (f, g)$ almost everywhere on $I$.