  Hara and Yoshida introduced a notion of $\aaa$-tight closure in 2003, and
they proved that the test ideals given by this operation correspond to
multiplier ideals. However, their operation is not a true closure. The
alternative operation introduced here is a true closure. Moreover, we define a
joint Hilbert-Kunz multiplicity that can be used to test for membership in this
closure. We study the connections between the Hara-Yoshida operation and the
one introduced here, primarily from the point of view of test ideals. We also
consider variants with positive real exponents.
