use bounded continuous functions that vanish outside one of the open sets This applies in particular when and when is the Lebesgue measure. The space of continuous and compactly supported functions is dense in Similarly, the space of integrable step functions is dense in this space is the linear span of indicator functions of bounded intervals when of bounded rectangles when and more generally of products of bounded intervals. Several properties of general functions in are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on in the following sense: where Closed subspaces If is a probability measure on a measurable space is any positive real number, and is a vector subspace, then is a closed subspace of if and only if is finite-dimensional (note that was chosen independent of ). In this theorem, which is due to Alexander Grothendieck, it is crucial that the vector space be a subset of since it is possible to construct an infinite-dimensional closed vector subspace of (that is even a subset of ), where is Lebesgue measure on the unit circle and is the probability measure that results from dividing it by its mass Let be a measure space. If then can be defined as above: it is the quotient vector space of those measurable functions such that As before, we may introduce the -norm but does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality valid for implies that and so the function is a metric on The resulting metric space is complete; the verification is similar to the familiar case when The balls form a local base at the origin for this topology, as ranges over the positive reals. These balls satisfy for all real which in particular shows that is a bounded neighborhood of the origin; in other words, this space is locally bounded, just like every normed space, despite not being a norm. In this setting satisfies a reverse Minkowski inequality, that is for This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces for . The space for is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is