Current induced by a locally integrable differential form: I don’t understand why it is not trivially 00

Today this question captured my attention, hence I want to generalize it.

Let XX be a complex manifold of dimension MM and let ω\omega be a (n−p,n−q)(n-p,n-q)-differential form such that in each chart it is represented by locally integrable functions. A standard example of current on XX (see for example De Rham – Differential manifolds, Chap. III example 2) is the following:

[ω]:α↦∫Xα∧ω[\omega]:\alpha \mapsto\int_X \alpha\wedge\omega
where α\alpha is a C∞C^\infty (p,q)(p,q)-form and α∧ω\alpha\wedge\omega is a locally integrable (n,n)(n,n)-form.

Who ensures that α∧ω\alpha\wedge\omega is integrable? Usually the integral is defined for smooth differential forms on oriented manifolds (see for example Lee’s book). About this point I’m quite sure the answer will be: “integration can be extended to locally integrable forms”, I just wanted to check.
Consider n=1n=1, p=0p=0, q=0q=0 and let’es examine the example of the question linked above:
ω=∂ˉ∂log|f|\omega=\partial\bar\partial\log|f|
where ff is a meromorphic function on the Riemann surface XX. Then the current [ω][\omega] is

[ω]:g↦∫Xgω[\omega]:g \mapsto\int_X g\omega

for any C∞C^\infty function gg. Here the problem: note that ω\omega is 00 almost everywhere, in particular ω\omega is supported in the set of zeroes and poles of ff i.e. in a finite set! Why is the integral ∫Xgω\int_X g\omega different from 00? It seems almost obvious to deduce that the integral of a differential form supported in a finite set is 00 because of the properties of the Riemann integral in Rn\mathbb R^n. I’ve been thinking to this fact all the day but without any solution.

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1 Answer
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Concerning the first bullet point: Yes, integration can be extended to non-smooth forms. Locally integrable isn’t quite sufficient, unless the manifold is compact. In your setting, one would typically require that α\alpha has compact support. Then α∧ω\alpha \wedge \omega is a locally integrable form with compact support, and hence integrable. One can also require fast enough decay for α\alpha and impose growth conditions for ω\omega – think e.g. of of the Schwartz space of rapidly decreasing functions and its dual, the space of tempered distributions; not every locally integrable function defines a tempered distribution, it mustn’t grow too rapidly at ∞\infty.

Concerning the second bullet point, we interpret log|f|\log \lvert f\rvert as a current, and hence don’t consider the classical derivative where it exists, but the distributional derivative (I’m not sure whether there’s a different term in the context of currents, but the idea is the same as with distributions and test functions). Thus when you integrate, you shift the differentiations over to gg via integration by parts. Evaluating such an integral then shows that [ω][\omega] is some linear combination of evaluation at the zeros and poles of ff (the coefficients depending on the order of the respective zeros and poles).