Lebesgue Integrals of Real Functions#

Definition: Lebesgue Integral

Let \((\mathbb{R}, \Sigma, \mu)\) be the measure space formed using the Lebesgue measure on \(\mathbb{R}\).

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a measurable real function on a Lebesgue-measurable subset \(\mathcal{D}\) and let \(S \subseteq \mathbb{R}\) also be Lebesgue-measurable.

The Lebesgue integral of \(f\) over \(S\) is the Lebesgue integral of \(f\) over \(S\) with respect to the Lebesgue measure \(\mu\):

\[ \int_S f \mathop{\mathrm{d}\mu} \]

Definition: Lebesgue-Integrability

We say that \(f\) is Lebesgue-integrable on \(S\) if its Lebesgue integral is finite.

Theorem: Riemann-Integrability \(\implies\) Lebesgue-Integrability

Let \(f: I \subset \mathbb{R} \to \mathbb{R}\) be a real function on a closed interval \(I = [a,b]\).

If \(f\) is Riemann-integrable on \(I\), then \(f\) is also Lebesgue-integrable on \(I\) with

\[ \int_a^b f(x) \mathop{\mathrm{d}x} = \int_I f \mathop{\mathrm{d}\mu} \]

Proof

TODO

Theorem: Equality of Lebesgue Integrals

Let \(f: \mathcal{D}_f \subseteq \mathbb{R} \to \mathbb{R}\) and \(g: \mathcal{D}_g \subseteq \mathbb{R} \to \mathbb{R}\) be measurable real functions on Lebesgue-measurable subsets \(\mathcal{D}_f\) and \(\mathcal{D}_g\) and let \(S \subseteq \mathbb{R}\) also be Lebesgue-measurable.

If \(f\) and \(g\) are non-negative on \(S\) and the set \(\{x \in S \mid f(x) \ne g(x)\}\) is a subset of a null set, then the Lebesgue integrals of \(f\) and \(g\) over \(S\) are equal.

\[ \int_S f \mathop{\mathrm{d}\mu} = \int_S g \mathop{\mathrm{d}\mu} \]

Proof

TODO

Theorem: Linearity of Lebesgue Integrals

Let \(f: \mathcal{D}_f \subseteq \mathbb{R} \to \mathbb{R}\) and \(g: \mathcal{D}_g \subseteq \mathbb{R} \to \mathbb{R}\) be measurable real functions on Lebesgue-measurable subsets \(\mathcal{D}_f\) and \(\mathcal{D}_g\) and let \(S \subseteq \mathbb{R}\) also be Lebesgue-measurable.

If \(f\) and \(g\) are Lebesgue-integrable on \(S\), then so is the function \(\alpha f + \beta g\) for all \(\alpha, \beta \in \mathbb{R}\). Furthermore,

\[ \int_S \alpha f + \beta g \mathop{\mathrm{d}\mu} = \alpha \int_S f \mathop{\mathrm{d}\mu} + \beta \int_S g \mathop{\mathrm{d}\mu} \]

Proof

TODO