of choice, this space is much bigger than except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of is Embeddings Colloquially, if then contains functions that are more locally singular, while elements of can be more spread out. Consider the Lebesgue measure on the half line A continuous function in might blow up near but must decay sufficiently fast toward infinity. On the other hand, continuous functions in need not decay at all but no blow-up is allowed. The precise technical result is the following. Suppose that Then: if and only if does not contain sets of finite but arbitrarily large measure, and if and only if does not contain sets of non-zero but arbitrarily small measure. Neither condition holds for the real line with the Lebesgue measure. In both cases the embedding is continuous, in that the identity operator is a bounded linear map from to in the first case, and to in the second. (This is a consequence of the closed graph theorem and properties of spaces.) Indeed, if the domain has finite measure, one can make the following explicit calculation using Hölder's inequality leading to The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity is precisely the case of equality being achieved exactly when -almost-everywhere. Dense subspaces Throughout this section we assume that Let be a measure space. An integrable simple function on is one of the form where are scalars, has finite measure and is the indicator function of the set for By construction of the integral, the vector space of integrable simple functions is dense in More can be said when is a normal topological space and its Borel –algebra, i.e., the smallest –algebra of subsets of containing the open sets. Suppose is an open set with It can be proved that for every Borel set contained in and for every there exist a closed set and an open set such that It follows that there exists a continuous Urysohn function on that is on and on with If can be covered by an increasing sequence of open sets that have finite measure, then the space of –integrable continuous functions is dense in More precisely, one can