Even with the Riemann Integral, we do not usually use the definition (as a limit of Riemann sums, or by verifying that the limit of the upper sums and the lower sums both exist and are equal) to compute integrals. Instead, we use the Fundamental Theorem of Calculus, or theorems about convergence. The following are taken from Frank E. Burk's A Garden of Integrals, which I recommend. One can use these theorems to compute integrals without having to go down all the way to the definition (when they are applicable).
Theorem (Theorem 3.8.1 in AGoI; Convergence for Riemann Integrable Functions) If $\{f_k\}$ is a sequence of Riemann integrable functions converging uniformly to the function $f$ on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$ and
$$R\int_a^b f(x)\,dx = \lim_{k\to\infty}R\int_a^b f_k(x)\,dx$$
(where "$R\int_a^b f(x)\,dx$" means "the Riemann integral of $f(x)$").
Theorem (Theorem 3.7.1 in AGoI; Fundamental Theorem of Calculus for the Riemann Integral) If $F$ is a differentiable function on $[a,b]$, and $F'$ is bounded and continuous almost everywhere on $[a,b]$, then:
- $F'$ is Riemann-integrable on $[a,b]$, and
- $\displaystyle R\int_a^x F'(t)\,dt = F(x) - F(a)$ for each $x\in [a,b]$.
Likewise, for Riemann-Stieltjes, we don't usually go by the definition; instead we try, as far as possible, to use theorems that tell us how to evaluate them. For example:
Theorem (Theorem 4.3.1 in AGoI) Suppose $f$ is continuous and $\phi$ is differentiable, with $\phi'$ being Riemann integrable on $[a,b]$. Then the Riemann-Stieltjes integral of $f$ with respect to $\phi$ exists, and
$$\text{R-S}\int_a^b f(x)d\phi(x) = R\int_a^b f(x)\phi'(x)\,dx$$
where $\text{R-S}\int_a^bf(x)d\phi(x)$ is the Riemann-Stieltjes integral of $f$ with respect to $d\phi(x)$.
Theorem (Theorem 4.3.2 in AGoI) Suppose $f$ and $\phi$ are bounded functions with no common discontinuities on the interval $[a,b]$, and that the Riemann-Stieltjes integral of $f$ with respect to $\phi$ exists. Then the Riemann-Stieltjes integral of $\phi$ with respect to $f$ exists, and
$$\text{R-S}\int_a^b \phi(x)df(x) = f(b)\phi(b) - f(a)\phi(a) - \text{R-S}\int_a^bf(x)d\phi(x).$$
Theorem. (Theorem 4.4.1 in AGoI; FTC for Riemann-Stieltjes Integrals) If $f$ is continuous on $[a,b]$ and $\phi$ is monotone increasing on $[a,b]$, then $$\displaystyle \text{R-S}\int_a^b f(x)d\phi(x)$$
exists. Defining a function $F$ on $[a,b]$ by
$$F(x) =\text{R-S}\int_a^x f(t)d\phi(t),$$
then
- $F$ is continuous at any point where $\phi$ is continuous; and
- $F$ is differentiable at each point where $\phi$ is differentiable (almost everywhere), and at such points $F'=f\phi'$.
Theorem. (Theorem 4.6.1 in AGoI; Convergence Theorem for the Riemann-Stieltjes integral.) Suppose $\{f_k\}$ is a sequence of continuous functions converging uniformly to $f$ on $[a,b]$ and that $\phi$ is monotone increasing on $[a,b]$. Then
The Riemann-Stieltjes integral of $f_k$ with respect to $\phi$ exists for all $k$; and
The Riemann-Stieltjes integral of $f$ with respect to $\phi$ exists; and
$\displaystyle \text{R-S}\int_a^b f(x)d\phi(x) = \lim_{k\to\infty} \text{R-S}\int_a^b f_k(x)d\phi(x)$.
One reason why one often restricts the Riemann-Stieltjes integral to $\phi$ of bounded variation is that every function of bounded variation is the difference of two monotone increasing functions, so we can apply theorems like the above when $\phi$ is of bounded variation.
For the Lebesgue integral, there are a lot of "convergence" theorems: theorems that relate the integral of a limit of functions with the limit of the integrals; these are very useful to compute integrals. Among them:
Theorem (Theorem 6.3.2 in AGoI) If $\{f_k\}$ is a monotone increasing sequence of nonnegative measurable functions converging pointwise to the function $f$ on $[a,b]$, then the Lebesgue integral of $f$ exists and
$$L\int_a^b fd\mu = \lim_{k\to\infty} L\int_a^b f_kd\mu.$$
Theorem (Lebesgue's Dominated Convergence Theorem; Theorem 6.3.3 in AGoI) Suppose $\{f_k\}$ is a sequence of Lebesgue integrable functions ($f_k$ measurable and $L\int_a^b|f_k|d\mu\lt\infty$ for all $k$) converging pointwise almost everywhere to $f$ on $[a,b]$. Let $g$ be a Lebesgue integrable function such that $|f_k|\leq g$ on $[a,b]$ for all $k$. Then $f$ is Lebesgue integrable on $[a,b]$ and
$$L\int_a^b fd\mu = \lim_{k\to\infty} L\int_a^b f_kd\mu.$$
Theorem (Theorem 6.4.2 in AGoI) If $F$ is a differentiable function, and the derivative $F'$ is bounded on the interval $[a,b]$, then $F'$ is Lebesgue integrable on $[a,b]$ and
$$L\int_a^x F'd\mu = F(x) - F(a)$$
for all $x$ in $[a,b]$.
Theorem (Theorem 6.4.3 in AGoI) If $F$ is absolutely continuous on $[a,b]$, then $F'$ is Lebesgue integrable and
$$L\int_a^x F'd\mu = F(x) - F(a),\qquad\text{for }x\text{ in }[a,b].$$
Theorem (Theorem 6.4.4 in AGoI) If $f$ is continuous and $\phi$ is absolutely continuous on an interval $[a,b]$, then the Riemann-Stieltjes integral of $f$ with respect to $\phi$ is the Lebesgue integral of $f\phi'$ on $[a,b]$:
$$\text{R-S}\int_a^b f(x)d\phi(x) = L\int_a^b f\phi'd\mu.$$
For Lebesgue-Stieltjes Integrals, you also have an FTC:
Theorem. (Theorem 7.7.1 in AGoI; FTC for Lebesgue-Stieltjes Integrals) If $g$ is a Lebesgue measurable function on $R$, $f$ is a nonnegative Lebesgue integrable function on $\mathbb{R}$, and $F(x) = L\int_{-\infty}^xd\mu$, then
- $F$ is bounded, monotone increasing, absolutely continuous, and differentiable almost everywhere with $F' = f$ almost everywhere;
- There is a Lebesgue-Stieltjes measure $\mu_f$ so that, for any Lebesgue measurable set $E$, $\mu_f(E) = L\int_E fd\mu$, and $\mu_f$ is absolutely continuous with respect to Lebesgue measure.
- $\displaystyle \text{L-S}\int_{\mathbb{R}} gd\mu_f = L\int_{\mathbb{R}}gfd\mu = L\int_{\mathbb{R}} gF'd\mu$.
The Henstock-Kurzweil integral likewise has monotone convergence theorems (if $\{f_k\}$ is a monotone sequence of H-K integrable functions that converge pointwise to $f$, then $f$ is H-K integrable if and only if the integrals of the $f_k$ are bounded, and in that case the integral of the limit equals the limit of the integrals); a dominated convergence theorem (very similar to Lebesgue's dominated convergence); an FTC that says that if $F$ is differentiable on $[a,b]$, then $F'$ is H-K integrable and
$$\text{H-K}\int_a^x F'(t)dt = F(x) - F(a);$$
(this holds if $F$ is continuous on $[a,b]$ and has at most countably many exceptional points on $[a,b]$ as well); and a "2nd FTC" theorem.
Best Answer
I think it is a bit of a shame that the standard pedagogical motivation for the Lebesgue integral seems to involve "dumping on" the Riemann integral.
There is (of course) a sense in which the Lebesgue integral is stronger: the collection of Lebesgue integrable functions properly contains the collection of (properly!) Riemann integrable functions, so the Lebesgue integral is "better".
As Mariano has pointed out in the comments, this is not necessarily very convincing: the standard examples of bounded, measurable, non-Riemann integrable functions look rather contrived.
In my opinion, most of the true advantage of the Lebesgue integral over the Riemann integral resides in the Dominated Convergence Theorem. This all-important result is much harder to prove directly for the Riemann integral. In part of course it is hard to prove because it is not true that a pointwise limit of Riemann integrable functions must be Riemann integrable, but again that's not where the crux of the problem lies. In the setting of the DCT if we add the hypothesis that the limit function is Riemann integrable then of course the theorem holds for the Riemann integral...but try to prove it without using Lebesgue's methods! (People have done this, by the way, and the difficulty of these arguments is persuasive evidence in favor of Lebesgue.)
I honestly think that in many (certainly not all, of course) areas of mathematics, it is the DCT (and a couple of other related results) which is really important and not the attendant measure theory at all. Thus I wish the approach via the Daniell integral were more popular: e.g. I can imagine an alternate universe in which this is part of undergraduate analysis and "measure theory and Lebesgue integration" was a popular "topics" graduate course rather than something that every young math student cuts her teeth on and many never use again. If measure theory were more divorced from the needs of integration theory one would naturally be tempted to either introduce more geometry or make explicit the connections to probability theory: either one of these would be a major livening up of the material, I think.
Right, but I'm meant to be answering the question rather than ranting.
There is another sense in which the Riemann integral is stronger than the Lebesgue integral: since Riemann's definition of Riemann integrability is a priori so demanding, knowing that a function is Riemann integrable is better than knowing it is Lebesgue integrable. It can be used to evaluate certain limits, yes, but this is not just a trick! Rather, the fact that an incredibly broad range of "interpolatory sums" associated to e.g. an arbitrary continuous function all converge to the same number is incredibly useful. As I have said before and others have said here, the entire branch of analysis known as approximation theory sure looks like it is founded upon the back of the Riemann integral, not the Lebesgue integral. In this branch of mathematics one is interested in various interpolatory schemes closely related to Riemann sums, and often one looks for a good tradeoff between convergence rates, efficiency and so forth in terms of the amount of smoothness of the function. An approximation scheme which worked for every $C^2$ function, for instance, would be regarded as quite general and useful. Does a numerical analyst ever meet a non-Riemann-integrable function?
It helps to fix ideas to restrict to the characteristic function $1_S$ of a bounded subset $S \subset \mathbb{R}^n$. Then $1_S$ is Lebesgue integrable iff $S$ is Lebesgue measurable. A general Lebesgue measurable set can be quite pathological. On the other hand, $1_S$ is Riemann integrable iff $S$ is Jordan measurable; this is a less well-known concept but is both technically useful and in some respects more natural. The fact that the volume of a Jordan measurable set can be computed as a limit of lattice-point counting is a key idea linking discrete and continous geometry. Just as an example, this came up (in a very standard and well-known way) in a paper I wrote recently: see Proposition 3.7 here. Geometric facts like these fail for, say, the characteristic function of the rational points in $[0,1]^d$.
Here is a somewhat related instance of Riemann integrability: a sequence $\{x_n\}$ in $[0,1]$ is uniformly distributed iff for all Riemann integrable functions $f: [0,1] \rightarrow \mathbb{R}$, $\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(x_n) = \int_0^1 f$. (See e.g. Theorem 7 of these notes.) On the right hand side it (of course) doesn't matter whether you take the integral to be in the sense of Riemann or Lebesgue, but if $f$ is not Riemann integrable then nothing good needs to happen on the left hand side. This is another instance in which Riemann integrable functions are better.