Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Definition

📜

The statement of the theorem

The theory of the Lebesgue integral requires a theory of measurable sets and measures on these sets, as well as a theory of measurable functions and integrals on these functions. Approximating a function by a simple function. One approach to constructing the Lebesgue integral is to make use of so-called simple functions: finite, real linear combinations of indicator functions. Simple functions that lie directly underneath a given function f can be constructed by partitioning the range of f into a finite number of layers. The intersection of the graph of f with a layer identifies a set of intervals in the domain of f, which, taken together, is defined to be the preimage of the lower bound of that layer, under the simple function. In this way, the partitioning of the range of f implies a partitioning of its domain. The integral of a simple function is found by summing, over these (not necessarily connected) subsets of the domain, the product of the measure of the subset and its image under the simple function (the lower bound of the corresponding layer); intuitively, this product is the sum of the areas of all bars of the same height. The integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions. To assign a value to the integral of the indicator function1_{S} of a measurable set S consistent with the given measureμ, the only reasonable choice is to set: ∫1S dμ=μ(S).\int 1_{S}\,d\mu =\mu (S). Notice that the result may be equal to +∞, unless μ is a finite measure. A finite linear combination of indicator functions ∑kak1Sk\sum _{k}a_{k}1_{S_{k}} where the coefficients a_{k} are real numbers and S_{k} are disjoint measurable sets, is called a measurable simple function. We extend the integral by linearity to non-negative measurable simple functions. When the coefficients a_{k} are positive, we set ∫(∑kak1Sk) dμ=∑kak∫1Sk dμ=∑kak μ(Sk)\int \left(\sum _{k}a_{k}1_{S_{k}}\right)\,d\mu =\sum _{k}a_{k}\int 1_{S_{k}}\,d\mu =\sum _{k}a_{k}\,\mu (S_{k}) whether this sum is finite or +∞. A simple function can be written in different ways as a linear combination of indicator functions, but the integral will be the same by the additivity of measures. Some care is needed when defining the integral of a real-valued simple function, to avoid the undefined expression ∞ − ∞: one assumes that the representation f=∑kak1Skf=\sum _{k}a_{k}1_{S_{k}} is such that μ(S_{k}) < ∞ whenever a_{k} ≠ 0. Then the above formula for the integral of f makes sense, and the result does not depend upon the particular representation of f satisfying the assumptions. (It is important that the representation be a finite linear combination, i.e. that k only take on a finite number of values.) If B is a measurable subset of E and s is a measurable simple function one defines ∫Bs dμ=∫1B s dμ=∑kak μ(Sk∩B).\int _{B}s\,\mathrm {d} \mu =\int 1_{B}\,s\,\mathrm {d} \mu =\sum _{k}a_{k}\,\mu (S_{k}\cap B). Let f be a non-negative measurable function on E, which we allow to attain the value +∞, in other words, f takes non-negative values in the extended real number line. We define ∫Ef dμ=sup⁡{ ∫Es dμ:0≤s≤f, s simple }.\int _{E}f\,d\mu =\sup \left\{\,\int _{E}s\,d\mu :0\leq s\leq f,\ s\ {\text{simple}}\,\right\}. We need to show this integral coincides with the preceding one, defined on the set of simple functions, when E is a segment [a, b]. There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is possible to prove that the answer to both questions is yes. We have defined the integral of f for any non-negative extended real-valued measurable function on E. For some functions, this integral ∫Ef dμ{\textstyle \int _{E}f\,d\mu } is infinite. It is often useful to have a particular sequence of simple functions that approximates the Lebesgue integral well (analogously to a Riemann sum). For a non-negative measurable function f, let s_{n}(x) be the simple function whose value is k/2^{n} whenever k/2^{n} ≤ f(x) < (k + 1)/2^{n}, for k a non-negative integer less than, say, 4^{n}. Then it can be proven directly that ∫f dμ=lim⁡n→∞∫sn dμ\int f\,d\mu =\lim _{n\to \infty }\int s_{n}\,d\mu and that the limit on the right hand side exists as an extended real number. This bridges the connection between the approach to the Lebesgue integral using simple functions, and the motivation for the Lebesgue integral using a partition of the range. To handle signed functions, we need a few more definitions. If f is a measurable function of the set E to the reals (including ±∞), then we can write f=f+−f−,f=f^{+}-f^{-}, where f+(x)={f(x)−if f(x)>0,0otherwise,f−(x)={−f(x)if f(x)<0,0otherwise.{\begin{aligned}f^{+}(x)&={\begin{cases}f(x){\hphantom {-}}&{\text{if }}f(x)>0,\\0&{\text{otherwise}},\end{cases}}\\f^{-}(x)&={\begin{cases}-f(x)&{\text{if }}f(x)<0,\\0&{\text{otherwise}}.\end{cases}}\end{aligned}} Note that both f^{+} and f^{−} are non-negative measurable functions. Also note that ∣f∣=f++f−.|f|=f^{+}+f^{-}. We say that the Lebesgue integral of the measurable function fexists, or is defined if at least one of ∫f+ dμ{\textstyle \int f^{+}\,d\mu } and ∫f− dμ{\textstyle \int f^{-}\,d\mu } is finite: min⁡(∫f+ dμ,∫f− dμ)<∞.\min \left(\int f^{+}\,d\mu ,\int f^{-}\,d\mu \right)<\infty . In this case we define ∫f dμ=∫f+ dμ−∫f− dμ.\int f\,d\mu =\int f^{+}\,d\mu -\int f^{-}\,d\mu . If ∫∣f∣ dμ<∞,\int |f|\,\mathrm {d} \mu <\infty , we say that f is Lebesgue integrable. That is, f belongs to the space L^{1}. It turns out that this definition gives the desirable properties of the integral. Assuming that f is measurable and non-negative, the function f∗(t) =def μ({x∈E∣f(x)>t}).f^{*}(t)\ {\stackrel {\text{def}}{=}}\ \mu \left(\{x\in E\mid f(x)>t\}\right). is monotonically non-increasing. The Lebesgue integral may then be defined as the improper Riemann integral of f^{∗}: ∫Ef dμ =def ∫0∞f∗(t) dt.\int _{E}f\,d\mu \ {\stackrel {\text{def}}{=}}\ \int _{0}^{\infty }f^{*}(t)\,dt. This integral is improper at the upper limit of ∞, and possibly also at zero. It exists, with the allowance that it may be infinite. As above, the integral of a Lebesgue integrable (not necessarily non-negative) function is defined by subtracting the integral of its positive and negative parts. Complex-valued functions can be similarly integrated, by considering the real part and the imaginary part separately. If h = f + ig for real-valued integrable functions f, g, then the integral of h is defined by ∫h dμ=∫f dμ+i∫g dμ.\int h\,d\mu =\int f\,d\mu +i\int g\,d\mu . The function is Lebesgue integrable if and only if its absolute value is Lebesgue integrable (see Absolutely integrable function). - ^This approach can be found in most treatments of measure and integration, such as Royden (1988). - ^Lemma 1 of page 76 of the second edition of Royden, Real Analysis. - ^ However, L^{1} is not "the space of Lebesgue integrable functions" but rather the space of equivalence classes of functions. - ^Lieb & Loss 2001 - ^If f^{∗} is infinite at an interior point of the domain, then the integral must be taken to be infinity. Otherwise f^{∗} is finite everywhere on (0, +∞), and hence bounded on every finite interval [a, b], where a > 0. Therefore the improper Riemann integral (whether finite or infinite) is well defined. - ^Equivalently, one could have defined f∗(t) = μ({x∈E∣f(x)≥t}),f^{*}(t)\ =\ \mu \left(\{x\in E\mid f(x)\geq t\}\right), since μ({x∈E∣f(x)≥t})=μ({x∈E∣f(x)>t})\mu \left(\{x\in E\mid f(x)\geq t\}\right)=\mu \left(\{x\in E\mid f(x)>t\}\right) for almost all ⁠ tt ⁠. - ^Rudin 1966