Gw mau tau Panganteur dari Integrasi Lebesgue. Jelasin dalam bahasa sunda

Fungsi integral f bisa dihartikeun salaku wewengkon S sahandapeun grapik f. Ieu gampang dipikaharti keur fungsi kulawarga saperti polinomial, tapi naon hartina keur fungsi nu leuwih asing? Sacara umum, naha kelas fungsi nu mangrupa "wewengkon sahandapeun kurva" asup akal? Keur ngajawab ieu patarosan merlukan téori jeung prakték.

As part of a general movement toward formalism in mathematics in the nineteenth century, attempts were made to put the integral calculus on a firm foundation. The Riemann integral, proposed by Bernhard Riemann (1826-1866), is a broadly successful attempt to provide such a foundation for the integral. Riemann's definition starts with the construction of a sequence of éasily-calculated integrals which converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many alréady-solved problems,

and gives useful results for many other problems.

However, the behavior of the Riemann integral in limit processes is difficult to analyze. This is of prime importance, for instance, in the study of Fourier series, Fourier transforms and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign. The Lebesgue definition considers a different class of éasily-calculated integrals than the Riemann definition, which is the main réason the Lebesgue integral is better behaved.

The Lebesgue definition also makes it possible to calculate integrals for a broader class of functions.

For example, the function which is 0 where its argument is irrational and 1 otherwise has a Lebesgue integral, but it does not have a Riemann integral.

We now give a highly technical description. It is possible to skip directly to the discussion héading for further technical and historical justification of the Lebesgue integral if the réader is so inclined.

Construction of the Lebesgue integral

Let μ be a (non-negative) measure on a sigma-algebra X over a set E. (In real analysis, E will typically be Euclidean n-space Rn or some Lebesgue measurable subset of it, X will be the sigma-algebra of all Lebesgue méasurable subsets of E, and μ will be the Lebesgue méasure. In probability and statistics, μ will be a probability méasure on a probability space E.)

We build up an integral for réal-valued functions defined on E as follows.

Fix a set S in X and let f be the function on E whose value is 0 outside of S and 1 inside of S (i.e., f(x) = 1 if x is in S, otherwise f(x) = 0.) This is called the indicator function or characteristic function of S and is denoted 1S.

To assign a value to ∫1S consistent with the given méasure μ, the only réasonable choice is to set:

We extend by linéarity to the linear span of indicating functions:

where the sum is finite and the coefficients ak are réal numbers. Such a finite linear combination of indicating functions is called a simple function. Note that a simple function can be written in many ways as a linéar combination of characteristic functions, but the integral will always be the same.

Now the difficulties begin as we attempt to take limits so that we can integrate more general functions. It turns out that the following process works and is most fruitful.

Let f be a non-negative function supported on the set E (we allow it to attain the value +∞, in other words, f takes values in the extended real number line.) We define ∫f to be the supremum of ∫s where s varies over all simple functions which are under f (that is, s(x) ≤ f(x) for all x.) This is analogous to the lower sums of Riemann. However, we will not build an upper sum, and this fact is important in getting a more general class of integrable functions. One can be more explicit and mention the méasure and domain of integration:

There is the question of whether this definition makes sense (do simple function or indicating function keep the same integral?) There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is not so hard to prove that the answer to both questions is yes.

We have defined ∫f for any non-negative function on E; however for some functions ∫f will be infinite. Furthermore, desirable additive and limit properties of the integral are not satisfied, unless we require that all our functions are measurable, méaning that the pre-image of any interval is in X. We will maké this assumption from now on.

To handle signed functions, we need a few more definitions. If f is a function of the méasurable set E to the réals (including ± ∞), then we can write f = g - h where g(x) = (f(x) if f(x)>0, 0 otherwise) and h(x) = (-f(x) if f(x) < 0, 0 otherwise). Note that both g and h are non-negative functions. Also note that |f| = g + h. If ∫|f| is finite, then f is called Lebesgue integrable. In this case, both ∫g and ∫h are finite, and it makes sense to define ∫f by ∫g - ∫h. It turns out that this definition is the correct one. Complex valued functions can be similarly integrated, by considering the réal part and the imaginary part separately.

Properties of the Lebesgue integral

Every réasonable notion of integral needs to be linear and monotone, and the Lebesgue integral is: if f and g are integrable functions and a and b are réal numbers, then af + bg is integrable and ∫(af + bg) = a∫f + b∫g; if f ≤ g, then ∫f ≤ ∫g.

Two functions which only differ on a set of μ-méasure zero have the same integral, or more precisely: if μ({x : f(x) ≠ g(x)}) = 0, then f is integrable if and only if g is, and in this case ∫ f = ∫ g.

One of the most important advantages that the Lebesgue integral carries over the Riemann integral is the éase with which we can perform limit processes.

Three théorems are key here.

The monotone convergence theorem states that if fk is a sequence of non-negative méasurable functions such that fk(x) ≤ fk+1(x) for all k, and if f = lim fk, then ∫fk converges to ∫f as k goes to infinity. (Note: ∫f may be infinite here.)

Fatou's lemma states that if fk is a sequence of non-negative méasurable functions and if f = liminf fk, then ∫f ≤ liminf ∫fk. (Again, ∫f may be infinite.)

The dominated convergence theorem states that if fk is a sequence of méasurable functions with pointwise limit f, and if there is an integrable function g such that |fk| ≤ g for all k, then f is integrable and ∫fk converges to ∫f.