Editing Lp space (section)

==={{math|''L''<sup>0</sup>}} space of measurable functions===

The vector space of ([[equivalence class]]es of) measurable functions on <math>(S, \Sigma, \mu)</math> is denoted <math>L^0(S, \Sigma, \mu)</math> {{harv|Kalton|Peck|Roberts|1984}}. By definition, it contains all the <math>L^p,</math> and is equipped with the topology of ''[[convergence in measure]]''. When <math>\mu</math> is a probability measure (i.e., <math>\mu(S) = 1</math>), this mode of convergence is named ''[[convergence in probability]]''. The space <math>L^0</math> is always a [[topological abelian group]] but  is only a [[topological vector space]] if <math>\mu(S)<\infty.</math> This is because scalar multiplication is continuous if and only if <math>\mu(S)<\infty.</math> If <math>(S,\Sigma,\mu)</math> is <math>\sigma</math>-finite then the [[weaker topology]] of [[local convergence in measure]] is an [[F-space]], i.e. a [[Complete topological vector space|completely]] [[metrizable topological vector space]]. Moreover, this topology is isometric to global convergence in measure <math>(S,\Sigma,\nu)</math> for a suitable choice of [[probability measure]] <math>\nu.</math>

The description is easier when <math>\mu</math> is finite. If <math>\mu</math> is a [[finite measure]] on <math>(S, \Sigma),</math> the <math>0</math> function admits for the convergence in measure the following [[fundamental system of neighborhoods]]
<math display="block">V_\varepsilon = \Bigl\{f : \mu \bigl(\{x : |f(x)| > \varepsilon\} \bigr) < \varepsilon \Bigr\}, \qquad \varepsilon > 0.</math>

The topology can be defined by any metric <math>d</math> of the form
<math display="block">d(f, g) = \int_S \varphi \bigl(|f(x) - g(x)|\bigr)\, \mathrm{d}\mu(x)</math>
where <math>\varphi</math> is bounded continuous concave and non-decreasing on <math>[0, \infty),</math> with <math>\varphi(0) = 0</math> and <math>\varphi(t) > 0</math> when <math>t > 0</math> (for example, <math>\varphi(t) = \min(t, 1).</math> Such a metric is called [[Paul Lévy (mathematician)|Lévy]]-metric for <math>L^0.</math> Under this metric the space <math>L^0</math> is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if <math>\mu(S)<\infty</math>. To see this, consider the Lebesgue measurable function <math>f:\mathbb R\rightarrow \mathbb R</math> defined by <math>f(x)=x</math>. Then clearly <math>\lim_{c\rightarrow 0}d(cf,0)=\infty</math>. The space <math>L^0</math> is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure <math>\lambda</math> on <math>\Reals^n,</math> the definition of the fundamental system of neighborhoods could be modified as follows
<math display="block">W_\varepsilon = \left\{f : \lambda \left(\left\{x : |f(x)| > \varepsilon \text{ and } |x| < \tfrac{1}{\varepsilon}\right\}\right) < \varepsilon\right\}</math>

The resulting space <math>L^0(\Reals^n, \lambda)</math>, with the topology of local convergence in measure, is isomorphic to the space <math>L^0(\Reals^n, g \, \lambda),</math> for any positive <math>\lambda</math>&ndash;integrable density <math>g.</math>