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.
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...
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...
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:
" ...
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:
" ...
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$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...
Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##,
$$\frac{\partial}{\partial r_k}\int_V...
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}...
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...
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...
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...
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...
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?
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...
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) =...
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...
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)
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...
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...
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!
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...
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...
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...
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...
I am looking for some good materials on Lebesgue integrals, especially anything with a geometric / visual flavor. Any suggestions would be greatly appreciated.