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}