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A weak and a strong concept of plurifinely plurisubharmonic and plurifinelyholomorphic functions are introduced. Strong will imply weak. The weak conceptis studied further. A function f is weakly plurifinely plurisubharmonic if andonly if f o h is finely subharmonic for all complex affine-linear maps h. As aconsequence, the regularization in the plurifine topology of a pointwisesupremum of such functions is weakly plurifinely plurisubharmonic, and itdiffers from the pointwise supremum at most on a pluripolar set. Weak plurifineplurisubharmonicity and weak plurifine holomorphy are preserved undercomposition with weakly plurifinely holomorphic maps.

Plurisubharmonic and holomorphic functions relative to the plurifine topology