5.6 Extending the definition of integrability

We now wish to extend the definition of the integral to a wider class of functions, namely to some that are unbounded and Others whose domains are not closed and bounded intervals.This extended definition is somewhat ad hoc, and these integrals are sometimes called “improper integrals.”

Let $f$ be a real or complex-valued function on the open interval $(a,b)$ where $a$ is possibly $-\infty$ and $b$ is possibly $+\infty .$ We say that $f$ is improperly-integrable on $(a,b)$ if it is integrable on each closed and bounded subinterval $[a^{\text{'}},b^{\text{'}}]\subset (a,b),$ and for each point $c\in (a,b)$ we have that the two limits $lim{b}^{\text{'}}\to b-0{\int }_c^{b^{\text{'}}}f$ and ${lim}_{a^{\text{'}}\to a+0}{\int }_{a^{\text{'}}}^{c}f$ exist.

More generally, We say that a real or complex-valued function $f,$ not necessarily defined on all of the open interval $(a,b),$ is improperly-integrable on $(a,b)$ if there exists a partition $\left\{x_i\right\}$ of $[a,b]$ such that $f$ is defined and improperly-integrable on each open interval $(x_{i-1},x_i).$

We denote the set of all functions $f$ that are improperly-integrable on an open interval $(a,b)$ by $I_i\left((a,b)\right).$

Analogous definitions are made for a function's being integrable on half-open intervals $[a,b)$ and $(a,b].$

Note that, in order for $f$ to be improperly-integrable on an open interval, we only require $f$ to be defined at almost all the points of the interval, i.e., at every point except the endpoints of some partition.

Part (a) of the preceding exercise is just the consistency condition we need in order to make a definition of the integral of an improperly-integrable function over an open interval.

Let $f$ be defined and improperly-integrable on an open interval $(a,b).$ We define the integral of $f$ over the interval $(a,b),$ and denote it by ${\int }_a^{b}f,$ by

$${\int }_a^{b}f=\underset{a^{\text{'}}\to a+0}{lim}{\int }_{a^{\text{'}}}^{c}f+\underset{b^{\text{'}}\to b-0}{lim}{\int }_c^{b^{\text{'}}}f.$$

In general, if $f$ is improperly-integrable over an open interval, i.e., $f$ is defined and improperly-integrable over each subinterval of $(a,b)$ determined by a partition $\left\{x_i\right\},$ then we define the integral of $f$ over the interval $(a,b)$ by

$${\int }_a^{b}f=\sum _{i=1}^n{\int }_{x_{i-1}}^{x_i}f.$$

Let $(a,b)$ be a fixed open interval (with $a$ possibly equal to $-\infty$ and $b$ possibly equal to $+\infty ),$ and let $I_i\left((a,b)\right)$ denote the set of improperly-integrable functions on $(a,b).$ Then:

1.   $I_i\left((a,b)\right)$ is a vector space of functions.
2.  (Linearity)  ${\int }_a^b(\alpha f+\beta g)=\alpha {\int }_a^{b}f+\beta {\int }_a^{b}g$ for all $f,g\in I_i\left((a,b)\right)$ and $\alpha ,\beta \in C.$
3.  (Positivity) If $f\left(x\right)\ge 0$ for all $x\in (a,b),$ then ${\int }_a^{b}f\ge 0.$
4.  (Order-preserving) If $f,g\in I_i\left((a,b)\right)$ and $f\left(x\right)\le g\left(x\right)$ for all $x\in (a,b),$ then ${\int }_a^{b}f\le {\int }_a^{b}g.$