The Lebesgue Integral

From Simple Functions to Dominated Convergence

Once we have established a rigorous theory for the measurement of sets, it is natural to begin the development of integration. We begin by defining the integral on the easiest case, simple functions, and use these results to build up to a more general integral.

Integral of Simple Functions

A simple function \(\phi(x)\) is defined as:

\phi(x) = \sum_{k = 1}^M c_k\chi_{E_k}(x)

This is a finite sum of constants multiplied by a characteristic function on a measurable subset \(E_k\). This is a generalization of the step functions in Riemann Sums; however, our sets \(E_k\) only need to be measurable, whereas Riemann limits them to intervals.

Definition: The Lebesgue integral of \(\phi(x)\) is:

\int_{\mathbb{R}^d}\phi(x) dx = \sum_{k = 1}^M c_k m(E_k)

Properties that hold for simple functions (and generalize later):

Linearity: \(\int(a\phi + b\psi) = a\int\phi + b\int\psi\)
Additivity: \(\int_{E\cup F}\phi = \int_{E}\phi + \int_{F}\phi\)
Monotonicity: \(\phi \leq \psi \implies \int\phi \leq \int\psi\)
Triangle Inequality: \(|\int \phi| \leq \int |\phi|\)

Bounded Functions & Finite Measure

A function is "supported on a set" \(E\) if it is non-zero only within \(E\). For a bounded function \(f\) supported on a set of finite measure, we use sequences of simple functions to define the integral.

Bounded Convergence Theorem

If \(\{\phi_n\}\) is a sequence of measurable functions bounded by \(M\), supported on a set \(E\) of finite measure, and \(\phi_n \to f\) a.e., then:

\lim_{n \to \infty} \int \phi_n = \int f

Integral of Nonnegative Functions

For a non-negative function \(f\), we define its integral as the supremum of the integrals of all measurable bounded functions \(g\) (supported on sets of finite measure) such that \(0 \leq g \leq f\).

Integrability: A measurable function \(f\) is integrable if \(\int_{\mathbb{R}^d} f(x)dx < \infty\).

If \(f\) is integrable, then \(f < \infty\) almost everywhere.
If \(\int f = 0\), then \(f = 0\) almost everywhere.

Fatou's Lemma & Monotone Convergence

We cannot always assume the limit of an integral equals the integral of the limit. Fatou's Lemma provides an essential lower bound.

Fatou's Lemma

For a sequence of non-negative measurable functions \(f_n \to f\) a.e.:

\int f \leq \liminf_{n \to \infty} \int f_n

Monotone Convergence Theorem (MCT)

If \(f_n\) is a sequence of non-negative measurable functions such that \(f_n \nearrow f\), then:

\lim_{n \to \infty} \int f_n = \int f

The Generalized Lebesgue Integral

To integrate any measurable function, we decompose it into its positive and negative parts: \(f = f^{+} - f^{-}\), where \(f^{+} = \max(f, 0)\) and \(f^{-} = \max(-f, 0)\).

\int f = \int f^{+} - \int f^{-}

A key property here is absolute continuity: If \(f\) is integrable, then \(\forall \varepsilon > 0\), there exists a \(\delta > 0\) such that if \(m(E) < \delta\), then \(\int_E |f| < \varepsilon\).

Dominated Convergence Theorem (DCT)

If \(f_n \to f\) a.e. and there exists an integrable function \(g\) such that \(|f_n| \leq g\) for all \(n\), then:

\lim_{n \to \infty} \int f_n = \int f

Conclusion

This framework forms the foundation of modern analysis. Further topics like \(L^p\) spaces, Fubini's Theorem, and convolutions build directly upon these convergence results.

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