The Riemann Integral, Part Two

Focus attention on a closed and bounded $k$-dimensional rectangle $I=[\boldsymbol {a}, \boldsymbol {b}]$. Suppose that $f$ and $g$ are RI on $I$. Using the main results of this section, quickly show that $af+bg$ [where $a, b \in \mathbb {R}$] and $fg$ are also RI. [The idea is to not regurgitate the calculus-based proofs. Observe by Claim 1 or Claim 2 (of this section) that $af+bg$ and $fg$ are continuous almost everywhere on $I$ if $f$ and $g$ are RI.]

Here, take $I=[0,1]$, and let $f$ denote the indicator function of the set of rational numbers in $[0,1]$.

Work the following parts:

Obtain $\int_{(0,1)} \frac{1}{\sqrt{x}} \, d\lambda (x)$ if it exists, otherwise show that it fails to exist. Observe that $\frac{1}{\sqrt{x}}$ is not bounded on $(0,1)$, so that one cannot immediately appeal to Claim 1 or Claim 2 (of this section).

$\int_{\mathbb {R}} \frac{1}{1+x^{2}} \, d\lambda (x)$ exists and equals $\pi$. [Be more careful here than merely stating that the improper Riemann integral $\int_{-\infty }^{+\infty } \frac{1}{1+x^{2}} \, dx$ exists and equals $\pi$.]

Is a RI function necessarily Borel measurable?

Show that $\int_{1}^{+\infty } \frac{1}{x} \, d\lambda (x)=+\infty$. [Be careful here.]

Let $f: \mathbb {R} \rightarrow \overline{\mathbb {R}}$ be such that $\int_{\mathbb {R}} f \, d\bar{\lambda }$ exists, is finite. Further suppose that $\int_{(a,b)} f \, d\bar{\lambda }=0$ for every $-\infty <a<b<+\infty$. Then $f=0$ $\bar{\lambda }$-a.e. on $\mathbb {R}$. [Work from indicators to simple functions, then to nonnegative functions, then to general integrable functions.]

Assume the result of Exercise 8 [problem 7.4.8 above]. Show that there cannot exist a set $E \in \mathcal{B}$ such that $\lambda \left(E \cap [a, b]\right)=\frac{1}{2}(b-a)$ for all real $a<b$. [Write the left side as $\int_{[a,b]} \; \mathbb {1}_{E} \, d\lambda$, and write the right side as $\int_{[a,b]} \frac{1}{2} \, d\lambda$. Use the previous exercise, working by contradiction.]

There exists a real valued $f$ on $[0,1]$ such that $f$ is continuous on $(0,1]$, $\lim_{\delta \rightarrow 0^{+}} \int_{[\delta , 1]} f \, d\bar{\lambda }$ exists and is finite, but $\int_{[0,1]} f \, d\bar{\lambda }$ is undefined.

Suppose that $f$ is improperly Riemann integrable over the real line, so that

exists as a real number. [Observe that this implies that $f$ is RI over each interval $[a, b]$.] Give sufficient conditions for $f$ so that $\int_{\mathbb {R}} f \, d\lambda =\int_{-\infty }^{+\infty } f(x) \, dx$.