REFERENCE SHEET

Lebesgue integration & Lᵖ spaces

Build integrals from measurable simple functions, then identify functions that differ only on null sets.

Before this σ-algebras & measures, Integration & accumulation

‖f‖ₚ = (∫ |f|ᵖ dμ)¹ᐟᵖ, 1≤p<∞

Definitions

Measurable function

For a real-valued f on (X,ℱ), preimages of Borel subsets of ℝ lie in ℱ.

Nonnegative integral

For f≥0 measurable, ∫f is the supremum of integrals of nonnegative simple functions s≤f. For integrable signed f, subtract the integrals of its positive and negative parts.

Lᵖ space

For 1≤p<∞, measurable functions with ∫|f|ᵖ<∞, modulo equality almost everywhere. L∞ uses the essential supremum norm.

Key theorems

Hölder inequality

If 1/p+1/q=1 with 1≤p,q≤∞, then ∫|fg|≤‖f‖ₚ‖g‖q for f∈Lᵖ and g∈Lq.

Minkowski inequality

For 1≤p≤∞, ‖f+g‖ₚ≤‖f‖ₚ+‖g‖ₚ.

Completeness

Lᵖ is a Banach space for 1≤p≤∞. L² is a Hilbert space with inner product given by integration of f times the complex conjugate of g.

Notable applications

Expectation

E[X] is the integral of a random variable against its probability measure.

Function approximation

L² error measures average squared discrepancy and supports orthogonal projection methods.

Work it out

Try before revealing

01 / Compute

For which finite p≥1 does f(x)=x⁻¹ᐟ² belong to Lᵖ(0,1) with Lebesgue measure?

Hint

Integrate x⁻ᵖᐟ² near zero.

Solution

The integral is finite exactly when p/2<1. Thus f∈Lᵖ for 1≤p<2, but not for p≥2.

02 / Explain

Are 1ℚ and 0 the same element of L²([0,1])?

Hint

They differ only on a null set.