What is Lebesgue integration: Definition and 24 Discussions

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the necessary abstractions for this.
The Lebesgue integral plays an important role in probability theory, real analysis, and many other fields in mathematics. It is named after Henri Lebesgue (1875–1941), who introduced the integral (Lebesgue 1904). It is also a pivotal part of the axiomatic theory of probability.
The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure.

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  1. cianfa72

    I ##L^2## square integrable function Hilbert space

    Hi, I'm aware of the ##L^2## space of square integrable functions is an Hilbert space. I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral...
  2. cianfa72

    A Karhunen–Loève theorem expansion random variables

    Hi, in the Karhunen–Loève theorem's statement the random variables in the expansion are given by $$Z_k = \int_a^b X_te_k(t) \: dt$$ ##X_t## is a zero-mean square-integrable stochastic process defined over a probability space ##(\Omega, F, P)## and indexed over a closed and bounded interval ##[a...
  3. Math Amateur

    MHB Lebesgue Integration of Simple Functions .... Lindstrom, Lemma 7.4.6 .... ....

    I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...
  4. Math Amateur

    I Lebesgue Integration of Simple Functions .... Lindstrom, Lemma 7.4.6 ...

    I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...
  5. DavideGenoa

    I Differentiating a particular integral (retarded potential)

    Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...
  6. DavideGenoa

    I Differentiation under the integral in retarded potentials

    Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##, $$\frac{\partial}{\partial r_k}\int_V...
  7. DavideGenoa

    I Laplacian of retarded potential

    Dear friends, I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by $$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...
  8. The_eToThe2iPi

    I Limitations of the Lebesgue Integral

    So I'm studying a course on measure theory and we've learned that the Lebesgue integral of a real function is (loosely) defined as the total area over the x-axis minus the total area under the x-axis. This seems to me to be limited because these areas can both be infinite but their difference...
  9. A

    Lebesgue integration over sets of measure zero

    Is it true in general that if f is Lebesgue integrable in a measure space (X,\mathcal M,\mu) with \mu a positive measure, \mu(X) = 1, and E \in \mathcal M satisfies \mu(E) = 0, then \int_E f d\mu = 0 This is one of those things I "knew" to be true yesterday, and the day before, and the...
  10. L

    Analysis Prelim prep: Lebesgue integration

    Hi everyone, I am studying past analysis prelim exams to take in the fall and have run into one which really has me stumped: Let f be a real-valued Lebesgue integral function on [0,\infty). Define F(x)=\int_{0}^{\infty}f(t)\cos(xt)\,dt. Show that F is defined on R and is continuous on R...
  11. B

    Very difficult Real Analysis question on Lebesgue integration

    Suppose gn are nonnegative and integrable on [0, 1], and that gn \rightarrow g almost everywhere. Further suppose that for all \epsilon > 0, \exists \delta > 0 such that for all A \subset [0, 1], we have meas(A) < \delta implies that supn \intA |gn| < \epsilon. Prove that g is integrable...
  12. W

    Lebesgue Integration: Finite Measure Not Sufficient

    Hello all, Here is my question: Suppose a measureable space (S,\mathcal{S},\mu) with \mu(S) < \infty and f : S \mapsto [0,\infty) , this is not yet sufficient to ensure \int_{S} f d \mu < \infty . Am I correct?
  13. W

    Implications of Lebesgue Integration for Bounded Functions

    Hello all, I am wondering the implication between almost everywhere bounded function and Lebesgue integrable. In the theory of Lebesgue integration, if a non-negative function f is bounded a.e., then it should be Lebesgue integrable, i.e. \int f d\mu < \infty because we do not take into...
  14. A

    Differentiation and Lebesgue integration

    Homework Statement Suppose g(x) = \int_0^x f(t) dt, where f is Lebesgue integrable on \mathbb R. Give an \epsilon - \delta proof that g'(y) = f(y) if y\in (0,\infty) is a point of continuity of f. Homework Equations The Attempt at a Solution I know I need to show that f(y) =...
  15. W

    Lebesgue Integration: Right-Continuous Function & Series Convergence

    Hello all, I would like to know when the Lebesgue integration w.r.t. a right-continuous function, there would be a series part which takes account of the jump components. Is it true that we require the series to be absolutely convergent? if so, what is the rationale of defining this instead of...
  16. A

    What values of p make \frac{1}{x^\alpha + x^\beta} integrable on (0,\infty)?

    I'm working through some old prelim problems, and one of them has me stumped: "For 0 < \alpha < \beta < \infty, for which positive real numbers p do we have \frac{1}{x^\alpha + x^\beta} \in L^p (0,\infty)
  17. B

    Green's function approach using Lebesgue integration

    I can't figure out how to use the Green's function approach rigorously, i.e., taking into account the fact that the Dirac Delta function is not a function on the reals. Suppose we have Laplace's Equation: \nabla^2 \phi(\vec{x})=f(\vec{x}) The solution, for "well-behaved" f(\vec{x}) is...
  18. I

    Limits and Lebesgue Integration

    Homework Statement Let (X,\Sigma,\mu) be a measure space. Suppose that {fn} is a sequence of nonnegative measurable functions, {fn} converges to f pointwise, and \int_X f = \lim\int_X f_n < \infty. Prove that \int_E f = \lim\int_E f_n for all E\in\Sigma. Show by example that this need not be...
  19. K

    Sequence of continuous functions vs. Lebesgue integration

    This is a question from Papa Rudin Chapter 2: Find continuous functions f_{n} : [0,1] -> [0,\infty) such that f_{n} (x) -> 0 for all x \in [0 ,1] as $n -> \infty. \int^{1}_{0} f_n dx -> 0 , but \int^{1}_{0} sup f_{n} dx = \infty. Any idea? :) Thank you so much!
  20. quasar987

    Solving Lebesgue Integration Problem on Dominated Convergence Theorem

    Homework Statement I have a HW sheet here on the dominated convergence theorem and this problem is giving me a hard time. It simply asks to show that \sum_{k=1}^{+\infty}\frac{1}{k^k}=\int_0^1\frac{dx}{x^x} The Attempt at a Solution Well, according the the cominated convergence thm...
  21. Oxymoron

    Understanding Lebesgue Integration

    Conventions X is a set. \mathcal{A} is a \sigma-algebra. Suppose that I have a measure space (X,\mathcal{A},\mu) and an \mathcal{A}-measurable function: f\,:\,X\rightarrow[0,\infty] All pretty regular stuff. Now, I have a "supposed" measure defined as \nu(E):=\int_E f\mbox{d}\mu for E\in...
  22. C

    How can we translate a lemma from Riemann integration to Lebesgue integration?

    Hello, I got a question about a lemma on Lebesgue integration (Riesz-Nagy approach). Let f(x) be a Lebesgue integrable function on interval (a, b). Riesz and Nagy (pg. 50 of Lessons of Functional Analysis) say that if f(x) is not bounded, for all epsilon > 0 we can decompose f(x) into the...
  23. C

    Ways of learning Lebesgue integration

    Good day. I am studying Lebesgue integration in Apostol’s Mathematical Analysis. I have learned already (I believe so) the Dominated Convergence Theorem and the Theorem of Differentiation under the integral sign. But Apostol does not introduce the Lebesgue integration by way of a Theory of...
  24. R

    Learn Geometric Lebesgue Integration | Suggestions Welcome

    I am looking for some good materials on Lebesgue integrals, especially anything with a geometric / visual flavor. Any suggestions would be greatly appreciated.