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I think that wath what you are getting at is called a "Current". See any book on geometric measure theory like the recent one by Krantz called "Geometric Integration Theory" or the older one by Federer called "Geometric Measure Theory"

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I think that wath you are getting at is called a "Current". See any book on geometric measure theory like the recent one by Krantz called "Geometric Integration Theory" or the older one by Federer called "Geometric Measure Theory"