Alternate definition for upper/lower integrals

Question

Suppose $f\colon [a,b] \to \mathbb R$ is a bounded function, and let $L(f,P)$ and $U(f,P)$ denote the lower and upper Darboux sums, respectively, of $f$ over the partition $P$. Consider $\{P_n\}_{n = 1}^\infty$, the sequence of regular partitions of $[a,b]$, where each $P_n$ has length $(b - a)/n$. We know that the upper and lower integrals are defined as $$\underline{\int_a^b}f \equiv \sup\{L(f,P),\text{$P$ is a partition of $[a,b]$}\}$$ and $$\overline{\int_a^b}f \equiv \inf\{U(f,P),\text{$P$ is a partition of $[a,b]$}\}.$$ However, I'm wondering if we can alternatively say that $$\underline{\int_a^b}f = \lim_{n \to \infty}L(f,P_n)$$ and $$\overline{\int_a^b}f = \lim_{n \to \infty}U(f,P_n).$$ My gut feeling is that this is true, because a regular partition can be a refinement of any other partition (by simply choosing the one of length less than or equal to the minimum interval length of the irregular partition) and then the rest follows, but I'm not sure how I could rigorously prove this, or even if this is true in the first place. Any hints/suggestions would be highly appreciated.

Answer 1

Since the upper integral is the greatest lower bound of upper sums, for any $\epsilon > 0$ there is a partition $P_\epsilon$ such that

$$\overline{\int_a^b f} \leqslant U(f,P_\epsilon) < \overline{\int_a^b f} + \epsilon/2.$$

Let $D = \sup\{|f(x)-f(y)|:x,y \in [a,b] \}$ denote the maximum oscillation of $f$ and let $\delta = \epsilon/(2mD)$ where $m$ is the number of points in the partition $P_\epsilon$.

Given an arbitrary partition $P = (x_0,x_1, \ldots, x_n)$, define the partition norm as

$$||P|| = \max_{1 \leqslant k \leqslant n}(x_k - x_{k-1}).$$

Now let $P$ be any partition with $||P||<\delta$. Form the common refinement $Q = P \cup P_\epsilon$. The difference between the upper sums $U(f,P)$ and $U(f,Q)$ has non-zero contributions from at most $m$ rectangles of width bounded by $\delta$ and height bounded by $D$.

Hence,

$$|U(f,P) - U(f,Q)| < m \cdot\frac{\epsilon}{2mD} \cdot D = \frac{\epsilon}{2},$$

Since $Q$ is a refinement of $P_\epsilon$, we have $U(Q,f) \leqslant U(P_\epsilon,f),$ and

$$\overline{\int_a^b f} \leqslant U(f,P) < U(f,Q) + \frac{\epsilon}{2} \leqslant U(f,P_\epsilon) + \frac{\epsilon}{2} < \overline{\int_a^b f} + \epsilon.$$

Now choose $N \in \mathbb{N}$ such that $||P_N|| < \delta.$ For any $n \geqslant N$ we have $||P_n|| < \delta$ and

$$\overline{\int_a^b f} \leqslant U(f,P_n) < \overline{\int_a^b f} + \epsilon.$$

Thus,

$$\lim_{n \to \infty} U(f,P_n) = \overline{\int_a^b f}.$$

A similar argument may be used for the lower integral.