  A weak and a strong concept of plurifinely plurisubharmonic and plurifinely
holomorphic functions are introduced. Strong will imply weak. The weak concept
is studied further. A function f is weakly plurifinely plurisubharmonic if and
only if f o h is finely subharmonic for all complex affine-linear maps h. As a
consequence, the regularization in the plurifine topology of a pointwise
supremum of such functions is weakly plurifinely plurisubharmonic, and it
differs from the pointwise supremum at most on a pluripolar set. Weak plurifine
plurisubharmonicity and weak plurifine holomorphy are preserved under
composition with weakly plurifinely holomorphic maps.
