Reduction to Riemann Integrals

3.5. Reduction to Riemann Integrals#

The next theorem tells us that we may replace the symbol $d α$ with $α^{'} (x) d x$ under some conditions.

Theorem 3.6

Suppose $f \in R (α)$ on $[a, b]$ , and $α$ has a continuous derivative on $[a, b]$ . Then $f α^{'} \in R$ on $[a, b]$ , and

\int_{a}^{b} f (x) d α (x) = \int_{a}^{b} f (x) α^{'} (x) d x

Proof. First, suppose $f$ is bounded by $M > 0$ , i.e.,

(3.9)#

| f (x) | \leq M \forall x \in [a, b]

Let $ε > 0$ be arbitrary.

Because $α^{'}$ is continuous on $[a, b]$ , it is continuous uniformly there. There exists $δ > 0$ such that

(3.10)#

| s - t | ⟹ | α^{'} (s) - α^{'} (t) | < \frac{ε}{2 M (b - a)}

Since $f$ is integrable w.r.t. $α$ on $[a, b]$ , there exists a partition $P_{ε}$ of $[a, b]$ such that for any refinement $P$ of $P_{ε}$ , and any list of representatives $T$ of $P$ , we have

(3.11)#

| S (P, T, f, α) - \int_{a}^{b} f d α | < ε / 2

Then, we can find a finer partition $P_{ε}^{'} \supseteq P_{ε}$ such that $‖ P_{ε}^{'} ‖ < δ$ .

Let $P \supseteq P_{ε}^{'}$ be a refinement such that and $T$ be a list of representatives of $P$ . Note that $P$ is of course also a refinement of $P_{ε}$ . Applying the mean value theorem, we have

\begin{aligned} S (P, T, f, α) & = \sum_{k = 1}^{n} f (t_{k}) [α (t_{k}) - α (t_{k - 1})] \\ = \sum_{k = 1}^{n} f (t_{k}) α^{'} (s_{k}) Δ x_{k} \end{aligned}

where each $s_{k} \in (x_{k - 1}, x_{k})$ .

Taking the difference of $S (P, T, f α^{'}, x)$ and $S (P, T, f, α)$ , we have

\begin{aligned} | S (P, T, f α^{'}, x) - S (P, T, f, α) | & = | \sum_{k = 1}^{n} f (t_{k}) [α^{'} (t_{k}) - α^{'} (s_{k})] Δ x_{k} | \\ \leq \sum_{k = 1}^{n} | f (t_{k}) [α^{'} (t_{k}) - α^{'} (s_{k})] Δ x_{k} | \\ = \sum_{k = 1}^{n} | f (t_{k}) | | α^{'} (t_{k}) - α^{'} (s_{k}) | Δ x_{k} \end{aligned}

Then applying (3.9) and (3.10), the above difference is further bounded by

(3.12)#

| S (P, T, f α^{'}, x) - S (P, T, f, α) | < M \frac{ε}{2 M (b - a)} \sum_{k = 1}^{n} Δ x_{k} = ε / 2

Recall $P \supseteq P_{ε}$ . Then we may conclude this proof by comparing (3.11) and (3.12).