Existence of almost Cohen-Macaulay algebras implies the existence of big Cohen-Macaulay algebras

In [1], the dagger closure is extended over finitely generated modules over Noetherian local domain (R,m) and it is proved to be a Dietz closure. In this short note we show that it also satisfies the 'Algebra axiom' of [9] and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it.