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:

\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}

- On the
**Linearity of Integrals of Step Functions on General Intervals**page we saw that the integrals of step funcitons have a linearity property. For $f, g \in S(I)$, additivity holds:

\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:

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

- Afterwards, on the
**A Comparison Theorem for Integrals of Step Functions on General Intervals**page we looked at a nice comparison theorem for integrals of step functions. We saw that if $f, g \in S(I)$ and $f(x) \leq g(x)$ for all $x \in I$ then:

\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.

- On the
**The Limit of the Integral of a Decreasing Sequence of Nonnegative Step Functions Approaching 0 a.e. on General Intervals**page we noted that a sequence of functions $(f_n(x))_{n=1}^{\infty}$ with common domain $I$ is said to be**Increasing**on $I$ if for all $n \in \mathbb{N}$ and for all $x \in I$ we have that:

\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:

\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:

\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:

\begin{align} \quad \int_I g(x) \: dx \leq \lim_{n \to \infty} \int_I f_n(x) \: dx \end{align}