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:

\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:

\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}

- On the
**Partial Linearity of Integrals of Upper Functions on General Interval**page we saw that most of the linearity properties of integrals of upper functions hold. If $f, g \in I(U)$ then:

\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:

\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 $U(I)$.

- On the
**A Comparison Theorem for Integrals of Upper Functions on General Intervals**page we looked at a nice comparison theorem for integrals of upper functions. We saw that if $f, g \in U(I)$ and $f(x) \leq g(x)$ almost everywhere on $I$ then:

\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:

\begin{align} \quad \int_I f(x) \: dx = \int_I g(x) \: dx \end{align}

- On the
**Additivity of Integrals of Upper Functions on Subintervals of General Intervals**page we proved that if $f \in U(I)$ and $I = I_1 \cup I_2$ where $I_1$ and $I_2$ intersect in at most one point and if $f(x) \geq 0$ almost everywhere on $I$, then $f \in U(I_1)$, $f \in U(I_2)$, and furthermore:

\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:

\begin{align} \quad \int_I f(x) \: dx = \int_{I_1} g(x) \: dx + \int_{I_2} h(x) \: dx \end{align}

- On the
**Riemann Integrable Functions as Upper Functions**page we proved that every Riemann integrable function is an upper function.

- 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:

\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:

\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 $I$.

- Lastly, on
**The Maximum and Minimum Functions as Upper Functions**page we saw that if $f$ and $g$ are upper functions on $I$ then $\max (f, g)$ and $\min (f, g)$ are also upper functions on $I$.