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The Lebesgue integral and convergence theorems

Integrating by slicing the range instead of the domain; monotone and dominated convergence.

Riemann slices the x-axis; Lebesgue slices the y-axis and asks how much of the domain sits at each height. The payoff is the convergence theorems: under mild conditions, the limit of the integrals is the integral of the limit. Picture it: horizontal strips under the curve instead of vertical ones. Think it: L¹ and L² become complete spaces — the Hilbert spaces of functional analysis.

Beispiel: `integrate e^(-x) dx from 0 to oo`

Calculus · Definite-Integrals

Integrate e^(-x) from 0 to oo

\int_{0}^{\infty} e^{- x}\, dx

Schritt für Schritt

\int_{0}^{\infty} e^{- x}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
u = - x,\quad du = -1\, dx
Substitute u = - x.
\int e^{- x}\, dx = \int - e^{u}\, d_u
Rewrite the integral in terms of u.
\int - e^{u}\, d_u = -1 \int e^{u}\, d_u
Pull the constant -1 out of the integral.
\int e^{u}\, d_u = e^{u}
∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).
= - e^{- x}
Substitute back u = - x.
F(\infty) - F(0) = \left(0\right) - \left(-1\right)
Fundamental theorem of calculus: plug in the limits.
= 1
Simplify.

Die Antwort aufzeigen

Symbols used here

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

Euler's number
2.71828…, the base whose exponential is its own derivative.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

B(x, r),\ \overline{A},\ \partial A

open ball, closure, boundary
Points within r of x; A plus its limit points; the edge of A.

\mu(A),\ \sigma\text{-algebra}

measure of A
Size of a set; the family of sets that can be measured.

\|f\|_p,\ \langle f, g \rangle

p-norm, inner product
Length of a function; the generalised dot product.

How to: The Lebesgue integral and convergence theorems

First find an antiderivative F, then evaluate F(b) − F(a).
Substitute u = - x.
Rewrite the integral in terms of u.
Pull the constant -1 out of the integral.
∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).
Substitute back u = - x.
Fundamental theorem of calculus: plug in the limits.
Simplify.

Questions people ask

What is wrong with the Riemann integral?

It fails on functions that oscillate too much, and it does not interact well with limits: the limit of integrable functions need not be integrable. Lebesgue's integral fixes both.

Versuch es selbst.

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Beispiel: integrate e^(-x) dx from 0 to oo