edit:
oops, the first version seems flawed. hence, after sleeping on it, i will make another attempt, but still without claim of sufficiency. here goes:
second attempt: CAVEAT: Throughout this discussion the word "integrable" means having a
finite integral.
As a non expert in lebesgue integration, I am nonetheless going to go out on a limb and suggest that lebesgue integration is not as scary as it seems. Yes the definitions and proofs are complicated to go through in detail, but the idea is simple enough. The idea of lebesgue integration is to build a larger space of integrable functions (larger than riemann integrable ones) in which limits of integrable functions are again integrable. moreover the elements of this space are just limits of riemann integrable functions. In particular, for those functions that are already riemann integrable, the lebesgue integral does not change. Thus the lebesgue integral of a riemann integrable function is the same as its riemann integral, hence is evaluated by the FTC, and hence the integral of a lebesgue integrable function is a limit of integrals obtained from the FTC. the only new feature is that there exist lebesgue integrable functions that are not riemann integrable, e.g. many unbounded functions.
thus you will only encounter the lebesgue integral essentially in two situations:
i) you have a function at hand which is not riemann integrable, and you want to know if it is lebesgue integrable, and then what its lebesgue integral is.
or else:
ii) you have a sequence of riemann integrable functions whose limit function is not riemann integrable, and you want to know whether that limit is lebesgue integrable, and then what that lebesgue integral equals.
the answer to all these questions is basically the same: namely, a function is a limit of riemann integrable functions, if and only if that limit is lebesgue integrable, and then the lebesgue integral of that limiting function is equal to the limit of the integrals of the riemann integrable functions approximating it.
so what do we mean by limit? this depends on a notion introduced by riemann but usually attributed to lebesgue by people who have not actually read riemann, which is apparently almost everyone. [I had the great good fortune some years back, to be assigned to review, for Math Reviews, the english translation of riemann's works, and hence was motivated, almost forced, to read riemann, aided enormously by the excellent translation. It took me something like one day per sentence to manage even a few pages.]
riemann, in his original paper defining his integral, also proved a theorem about exactly which functions are riemann integrable. namely a bounded function is "riemann integrable" (not his terminology) if and only if, for every e>0, the set of discontinuities of that function can be covered by an infinite sequence of intervals the (infinite) sum of whose lengths is < e, i.e. iff the set of discontinuities has "measure zero".
[the reason riemann is not credited with this result may also be partly that he proved a slightly different but trivially equivalent version of it, in his paper: "On the representation of a function by trigonometric series", chapter 5. I.e. he defined the "oscillation" of a function to be a positive number which is zero exactly at points of continuity, and showed that a function is riemann integrable if and only if for every d>0 and every e>0, the set of points with oscillation > d has
content < e, i.e. can be covered by a finite sequence of intervals of total length < e. Taking the union over all d = 1/n, for all n, and using compactness and the trivial fact that a countably infinite union of sets of content zero has measure zero, the lebesgue criterion immediately follows. Thus I am confident that lebesgue himself knew he was only rephrasing the result of riemann. ]
a set with the property given above for the set of discontinuities of a riemann integrable function is therefore now called a set of "(lebesgue) measure zero", and this criterion is called "lebesgue's criterion".
then the notion of limit that we want is essentially this: a sequence {fn} of functions converges to a function f, if and only if (they have the same domain and) the set of points x in the domain where {fn(x)} does not converge to f(x), has measure zero. otherwise stated: {fn} converges to f "almost everywhere". We also need a "cauchy" criterion on the sequence {fn} in terms of their (riemann) integrals.
thus f is lebegue integrable if and only if there is a sequence {fn) of riemann integrable functions (or smooth functions, or step functions) such that:
1) {fn} converges to f pointwise almost everywhere, and
2) the sequence {fn} is a cauchy sequence of functions, in the sense that the integrals of the differences {|fn-fm|} approach zero as n,m approach infinity.
In this case, the integrals of the {fn} form a cauchy, hence convergent, sequence of real numbers, and the integral of f is the limit of the integrals of the {fn}.
If you are familiar with metric spaces and the practice of enlarging a metric space to a complete metric space, the lebesgue integrable functions are exactly the enlargement of the metric space of riemann integrable functions to a complete metric space, in which the riemann integrable functions are dense. The metric, or norm, since it is a vector space, is that the (semi-) norm of a function is the integral of its absolute value.
One modification is that we have to introduce an equivalence relation on functions, where a lebesgue integrable function f is considered zero if the integral of |f| is zero, (i.e. if the norm of f is zero), if and only if the function f equals zero almost everywhere. In particular two lebesgue integrable functions are considered equal if and only if they differ only on a set of measure zero.
Moreover the two properties 1), 2), above are almost equivalent, in the following sense:
If f is any function which is almost everywhere the pointwise limit of a sequence {fn} of riemann integrable functions as in property 1),
and if you know all the functions |fn| are bounded above by some riemann or lebesgue integrable function g, then f is lebegue integrable, the sequence {fn} is cauchy in the sense of property 2), and the lebesgue integral of f equals the limit of the integrals of the {fn}. ("Dominated convergence")
Moreover, assume you have a sequence {fn} of riemann integrable functions, which is cauchy in the sense of property 2). Then at least some
subsequence of the sequence {fn} converges pointwise to some function f almost everywhere, in the sense of property 1). And in fact the subsequence can be chosen so that for every e>0, the convergence is uniform off a set which can be covered by a sequence of intervals of total length < e. The function f is, almost everywhere, independent of the choice of subsequence used to define it.
My original condition 2), which merely asked for the integrals of the {fn} to be a cauchy sequence of real numbers, is in fact sufficient for
monotone sequences. I.e. if {fn} is a monotone (say increasing) sequence of riemann integrable (or lebesgue integrable) functions whose (hence also monotone) sequence of integrals is a cauchy, i.e. bounded, sequence of real numbers, then the sequence of functions {fn} is cauchy in the stronger sense of the current property 2) above,
and converges almost everywhere pointwise to some lebesgue integrable function f. Moreover, the convergence is also true in the integral norm, i.e. the sequence of integrals of the differences {|f-fn|} converges to zero. ("Monotone convergence")
So lebesgue integration is nothing but the theory of limits of riemann integrals, and lebesgue - integrable functions are essentially pointwise limits (almost everywhere) of riemann integrable functions, with an extra condition so that the limit of the integrals makes sense.
There are also good Fubini theorems: i.e. a function which is integrable as a function of two variables can be integrated by repeated integration, and a function of two variables which is at least a pointwise limit almost everywhere of integrable functions, and for which the repeated integrals make sense, is itself integrable as a function of two variables, and hence the integral can be computed as a repeated integral. [I mention this since I once had a course from an expert who remarked (roughly) "if you have a problem in integration theory that won't yield either to dominated convergence or Fubini, you're probably in trouble".]
Here is an example (in one variable): let f be defined on [0,1] and have value 0 at all irrational points, and value n at rational points of form m/n in lowest terms. This function is unbounded on every interval, and discontinuous everywhere, hence nowhere near riemann integrable. However, since a countable set has measure zero, hence the rationals have measure zero, this function is equivalent in the sense above to the constant function zero, i.e. they differ only on a set of measure zero. Hence the lebesgue integral of f equals the integral of the zero function, namely zero. More generally, if your function differs from a riemann integrable function only on a set of measure zero, then your function has the same lebesgue integral as that riemann integrable function.
here is another one: the function x^(-1/2) on [0,1], but with value 0 at x=0. This cannot be made equal to a riemann integrable function by changing it only on a set of measure zero, but it is a pointwise limit of monotone increasing riemann integrable functions whose integrals converge, as in the theory of "improper integrals". So this function is lebesgue integrable and its lebesgue integral equals its value as an improper riemann integral.
Unfortunately I am not an expert, so I might again make some, possibly serious, mistakes here, but this in my opinion is the key idea. A nice clear place to read about it is in the great book "Functional Analysis" by Riesz-Nagy. I do not especially recommend Dieudonne's own book, (vol. 2 of his admittedly great Foundations of analysis), where to me at least it seems far more abstract. Indeed I take with more than a grain of salt his famous diatribe against riemann integration, which I consider somewhat misleading.
Here is a nice result from Riesz-Nagy (p.36) to show the clarity of their exposition: Suppose {fn} is a sequence of riemann-integrable (or lebesgue-integrable) functions such that the integrals of the absolute values |fn| form a convergent
series of real numbers; then the series whose terms are the functions fn, itself converges pointwise almost everywhere to a lebesgue integrable function f, and the series may be integrated term by term.
Moreover I believe Riesz-Nagy show that every lebesgue - integrable function is obtained either as a limit almost everywhere of a monotone sequence of ordinary step functions with bounded integrals (as in the monotone convergence theorem above), or as a difference of two such limits.
Experts I know in this subject seem to me mostly to enjoy the abstract theory of measure, and measurable sets, measurable functions, and like to develop everything from scratch, without taking any advantage of the theory of riemann integration. To be sure Dieudonne' does not do this, but his version is to me even more abstract, defining lebesgue - integrable functions as limits of smooth functions, somewhat as I do here, but in a way that looked to me a bit forbidding. Of course it might look clearer to me now, (indeed I believe he generalizes in some sense the Riesz-Nagy approach), but I cannot look back there now, since I rashly gave away most of my set of Dieudonne' when moving.
Again, since I am not expert, you will do yourself a favor by consulting an expert source. Books I like include Riesz-Nagy (Functional analysis), or Berberian (Fundamentals of real analysis) , or Lang (Analysis II), or Wendell Fleming (Functions of several variables), or (I don't have this last one, but some expert friends recommend it): Wheeden and Zygmund (Measure and Integral). Many people also like especially part I of the classic Real analysis, by H.L. Royden. These books may be harder to read than this sketch, but the advantage is that what they say will actually be correct, and will include proofs and (counter) examples. Oh yes, the book Counterexamples in analysis, by Gelbaum and Olmsted is fun too.
Here is an article going at things in the opposite order to here. I.e. after having developed lebesgue integration and measure theory in the usual way, he proves afterwards that all lebesgue integrable functions are limits of continuous functions in the integral norm, in particular they are all limits of riemann integrable functions.
https://www.math.utah.edu/~savin/L3_5210.pdf
Hence he could have defined lebesgue integrable functions as such limits.
