I find it is hard to catch the current, sometimes it is just the picture as its support set (if I do not miss it). What is the heart idea of the current? What are the benefits to introduce such an odd concept which make the theory more and more complicated?
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In case you are still interested in an answer, I encourage you to look at Leon Simon's notes. The idea of currents, is to serve as a far reaching generalization of oriented submanifolds. They were introduced as part of the fairly long and involved program that went into solving the Plateau's problem: such a minimization problem would require three important ingredients: a notion of convergence (topology), admissibility of the "limit", and finally lower semicontinuity. It turns out that these three are in some sense opposing forces, and currents provide the right language to meet all these three ingredients.