Exercise 7.2.6

Question

A tagged partition $\left(P,\left\{c_{k}\right\}\right)$ is one where in addition to a partition $P$ we choose a sampling point $c_{k}$ in each of the subintervals $\left[x_{k-1}, x_{k}\right]$. The corresponding Riemann sum, $$ R(f, P)=\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x_{k}, $$ is discussed in Section 7.1, where the following definition is alluded to. {\bf Riemann's Original Definition of the Integral}: A bounded function $f$ is integrable on $[a, b]$ with $\int_{a}^{b} f=A$ if for all $\epsilon>0$ there exists a $\delta>0$ such that for any tagged partition $\left(P,\left\{c_{k}\right\}\right)$ satisfying $\Delta x_{k}<\delta$ for all $k$, it follows that $$ |R(f, P)-A|<\epsilon . $$ Show that if $f$ satisfies Riemann's definition above, then $f$ is integrable in the sense of Definition 7.2.7. (The full equivalence of these two characterizations of integrability is proved in Section 8.1.)

Ulisse Mini · Accepted Answer

Let $\epsilon > 0$. By Riemann's definition we can easily form a partition $P_n$, and tag it to form $P_u$ with the sampling point close to the supremum, so that
$$|R(f,P_u) - A| < \epsilon/4 \text{ and } |U(f,P_n) - R(f,P_u)| < \epsilon/4$$
 Similarly we can retag the partition to form $P_l$ with the sampling point close to the infimum, to get
$$|R(f, P_l) - A| < \epsilon/4\text{ and }|L(f,P_n) - R(f, P_l)|<\epsilon/4$$
Applying the Triangle Inequality we get $U(f,P_n) - L(f,P_n) < \epsilon$, so by Theorem 7.2.8 $f$ is integrable.

Exercise 7.2.6

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Exercise 7.2.6

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