the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in or every open convex set containing the function is unbounded for the -quasi-norm; therefore, the vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space contains an infinite family of disjoint measurable sets of finite positive measure. The only nonempty convex open set in is the entire space . As a particular consequence, there are no nonzero continuous linear functionals on the continuous dual space is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space ), the bounded linear functionals on are exactly those that are bounded on namely those given by sequences in Although does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology. The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on rather than work with for it is common to work with the Hardy space whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the Hahn–Banach theorem still fails in for . , the space of measurable functions The vector space of (equivalence classes of) measurable functions on is denoted . By definition, it contains all the and is equipped with the topology of convergence in measure. When is a probability measure (i.e., ), this mode of convergence is named convergence in probability. The description is easier when is finite. If is a finite measure on the function admits for the convergence in measure the following fundamental system of neighborhoods The topology can be defined by any metric of the form where is bounded continuous concave and non-decreasing on with and when (for example, Such a metric is called Lévy-metric for Under this metric the space is complete (it is again an F-space). The space is in general not locally bounded, and not locally convex. For the infinite Lebesgue measure on the definition of the fundamental system of neighborhoods could be modified as follows The resulting space coincides as topological vector space with for any positive –integrable density Generalizations and extensions Weak Let be a measure space, and a measurable function with real or complex values on The distribution