Analysis MOC

Lebesgue space

Let (𝑋,Σ,𝜇) be a measure space and let 𝑝 ∈[1,∞]. The seminormed Lebesgue space L𝑝(𝑋,𝜇) is defined as the set of all measurable functions 𝑓 :𝑋 →ℂ with finite 𝑝-seminorm given by Lebesgue integral fun

‖𝑓‖𝑝=(∫𝑋|𝑓(𝑥)|𝑝𝑑𝜇(𝑥))1/𝑝<∞

where in the case of 𝑝 =∞ (assuming 𝜇(𝑋) ≠0) we get the essential supremum

‖𝑓‖∞=inf{𝐶∈ℝ≥0:𝜇({𝑠∈𝑋:|𝑓(𝑠)|>𝐶})=0}

The Lebesgue space 𝐿𝑝(𝑋,𝜇) is a Banach space given by the normed quotient L𝑝(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

proof

In case 𝑋 =ℕ and 𝜇 is the counting measure, one recovers Lebesgue sequence space. The special case of L2 space can be be endowed with the structure of a Hilbert space (see below)

Properties

• Lebesgue space forms an inner product space iff p=2

Alternate approach

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

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