Lebesgue space

Let be a measure space and let . The seminormed Lebesgue space is defined as the set of all measurable functions with finite -seminorm given by Lebesgue integral #m/def/anal/fun

where in the case of (assuming ) we get the essential supremum

The Lebesgue space is a Banach space given by the normed quotient , 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 and is the counting measure, one recovers space. The special case of space can be be endowed with the structure of a Hilbert space (see below)

Properties

forms an inner product space iff

Alternate approach

In the case an alternate approach is followed by Lyle Noakes, where one first defines with integration given by the Riemann integral, and then moving to the Banach completion which is defined as .

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