Property A For Metric Measure Space

Guoliang Yu has introduced a property A on discrete metric spaces which has many nice properties and important applications to the Novikov conjecture and the coarse Baum-Connes conjecture. R.Tessera has given a definition of property A and a quantitative property A for metric measure spaces. The aim of this paper is to study the relationship between these two property As and whether the property A for metric measure space has the similar nice properties. By discussing its coarsely dense discrete subspace and constructing a function, this paper obtains the following main result:a bounded geometry metric measure space has property A if and only if there exists some coarsely dense discrete subspace having Yu's property A. Thus the property A for metric measure space has properties similar to Yu's property A. Meanwhile, this paper shows a quantitative version of property A with unbounded A-profile having property A.