Charges, Currents and Distributions: Terminology clarification.

Question

I am trying to learn some measure theory by myself these days and, while reading some real analysis books I realize that some of the terminology is somehow related to electricity. For examples Charges, Currents and Distributions. Can somebody please explain me why this is so and what is the actual mathematical relationship between theses concepts. Thank you in advance for you consideration.

Answer 1

They are so-named and based on physics terminology by analogy.

For charges, or as they are more well-known by, signed measures, this is a generalization of the standard concept of measure. Measure assigns positive real numbers to certain subsets of $\mathbb{R}$. Signed measures are thus a generalization that assigns both positive and negative real numbers to certain subsets of $\mathbb{R}$, which can be viewed as the $(+)$ and $(-)$ charges in physics.

For distributions, I don't know if this was the original intent but one physics consequence is that it allows us to talk rigorously about such "functions" like the Dirac delta function that comes from physics.

For currents, look at this link on MathOverflow, where a similar question was answered, and the physics analogy there seems to be electrical currents in 3-dimensional space.