Integrals of Step Functions on General Intervals Review

# Integrals of Step Functions on General Intervals Review

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

• Recall from the Step Functions on General Intervals page that we generalized the concept of a step function. We said that a function $f$ is a Step Function on the interval $I$ if there exists a closed and bounded interval $[a, b] \subseteq I$ such that $f$ is a step function in the usual sense on $[a, b]$ (i.e., there exists a partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ such that $f$ is constant on each open subinterval $(x_{k-1}, x_k)$ for all $k \in \{1, 2, ..., n \}$) and such that $f(x) = 0$ for all $x \in I \setminus [a, b]$.
• Hence, a step function can be defined on any type of interval - open, closed, half-open/closed, unbounded, bounded, etc… Furthermore, the Set of All Step Functions on $I$ is denoted $S(I)$.
• On the Integrals of Step Functions on General Intervals page we noted that if $f$ is a step function on $I$, where $f(x) = c_k$ for all $x \in (x_{k-1}, x_k)$, for all $k \in \{1, 2, ..., n \}$ and $f(x) = 0$ for all $x \in I \setminus [a, b]$ then we already know that the integral of this step function on $I$ is:
(1)
\begin{align} \quad \int_I f(x) \: dx &= \int_a^b f(x) \: dx \\ \quad &= \sum_{k=1}^{n} c_k[x_k - x_{k-1}] \end{align}
(2)
\begin{align} \quad \int_I [f(x) + g(x)] \: dx = \int_I f(x) \: dx + \int_I g(x) \:dx \end{align}
• Similarly, for any $k \in \mathbb{R}$, homogeneity holds:
(3)
\begin{align} \quad \int_I kf(x) \: dx = k \int_I f(x) \: dx \end{align}
(4)
\begin{align} \quad \int_I f(x) \: dx \leq \int_I g(x) \: dx \end{align}
• On the Properties That Hold Almost Everywhere page, said that a property is said to hold Almost Everywhere on a set $D \subseteq \mathbb{R}$ if there exists a subset $S \subset M$ with $m(S) = 0$ such that the property holds for all $x \in D \setminus S$. Such terminology will be important later on.
(5)
\begin{align} \quad f_n(x) \leq f_{n+1}(x) \end{align}
• Similarly, we said that this sequence of functions is Decreasing on $I$ if for all $n \in \mathbb{N}$ and for all $x \in I$ we have that:
(6)
\begin{align} \quad f_n(x) \geq f_{n+1}(x) \end{align}
• We then looked at a critically important theorem that seemed intuitively obvious at face value. We saw that if $(f_n(x))_{n=1}^{\infty}$ is a decreasing sequence of nonnegative step functions that converges to $0$ almost everywhere on an interval $I$ then:
(7)
\begin{align} \quad \lim_{n \to \infty} \int_I f_n(x) \: dx = 0 \end{align}
• As a result of this theorem, we saw on the Another Comparison Theorem for Integrals of Step Functions on General Intervals page that if $(f_n(x))_{n=1}^{\infty}$ is an increasing sequence of step functions that converges to $f$ almost everywhere on $I$ and if $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ exists then for any step function $g$ such that $g(x) \leq f(x)$ almost everywhere on $I$ we have that:
(8)
\begin{align} \quad \int_I g(x) \: dx \leq \lim_{n \to \infty} \int_I f_n(x) \: dx \end{align}