Lebesgue space

Let $(𝑋, Σ, 𝜇)$ be a measure space and let $𝑝 \in [1, \infty]$ . The seminormed Lebesgue space $L 𝑝 (𝑋, 𝜇)$ is defined as the set of all measurable functions $𝑓 : 𝑋 \to ℂ$ with finite $𝑝$ -seminorm given by Lebesgue integral #m/def/anal/fun

‖ 𝑓 ‖ 𝑝 = (\int 𝑋 | 𝑓 (𝑥) | 𝑝 𝑑 𝜇 (𝑥)) 1 / 𝑝 < \infty

where in the case of $𝑝 = \infty$ (assuming $𝜇 (𝑋) \neq 0$ ) we get the essential supremum

‖ 𝑓 ‖ \infty = i n f {𝐶 \in ℝ \geq 0 : 𝜇 ({𝑠 \in 𝑋 : | 𝑓 (𝑠) | > 𝐶}) = 0}

The Lebesgue space $𝐿 𝑝 (𝑋, 𝜇)$ is a Banach space given by the normed quotient $L 𝑝 (𝑋, 𝜇)$ , whose elements are functions up to equality almost everywhere.

Proof of (semi)norm

Absolute homogeneity is immediately clear, while the the triangle inequality is given by the Minkowski inequality.

Proof of Banach space

#missing/proof

In case $𝑋 = ℕ$ and $𝜇$ is the counting measure, one recovers $ℓ 𝑝$ space. The special case of $𝐿 2$ space can be be endowed with the structure of a Hilbert space (see below)

Properties

$𝐿 𝑝 (𝑋, 𝜇)$ forms an inner product space iff $𝑝 = 2$

Alternate approach

In the case $𝑋 = [𝑎, 𝑏] \subseteq ℝ$ an alternate approach is followed by Lyle Noakes, where one first defines $˜ 𝐿 𝑝 (𝑋) = 𝐶 [𝑎, 𝑏]$ with integration given by the Riemann integral, and then moving to the Banach completion which is defined as $𝐿 𝑝 [𝑎, 𝑏]$ .

#state/tidy | #lang/en | #SemBr