3 Other Non-Bolognese Spaces

The results in this chapter are all borrowed from [ and none of them has been formalised during the workshop.

Theorem 3.1

Let \((X,\mu )\) be a measure space. For every \(0{\lt}p\) the space \(L^p(\mu )\) is a metric space with metric

\[ d(f, g) = \begin{cases} \left(\int _{X}|f-g|^{p} \mathrm{~ d} \mu \right)^{1 / p}, & \text{ if } p \geq 1 \\ \int _{X}|f-g|^{p} \mathrm{~ d} \mu , & \text{ if } 0{\lt}p{\lt}1 \end{cases} \]

Can Theorem 2.9 be generalised beyond the case \(X=[0,1]\)? The answer relies on the following

Theorem 3.2

For every \(0{\lt}p{\lt}1\), the space \(L^p(\mu )\) is Bolognese if and only if \(\mu \) assumes finitely many values.

As a corollary of the above result, we obtain a new proof of Theorem 2.9:

Corollary 3.3

The space \(L^{p}([0,1])\) of equivalence classes of measurable functions \(f\colon [0,1]\to \mathbb {R}\) satisfying \(\int _0^1\lvert f(x)\rvert ^{p}\mathrm{d}x {\lt} \infty \) (with respect to the Lebesgue measure) is not Bolognese.

Proof ▶