A modern take that is highly recommended for those interested in the "Isoperimetric Problem." Conclusion
These are sets that, while not necessarily smooth manifolds, can be covered by a countable collection of Lipschitz images of Euclidean space. They behave "almost" like manifolds.
Due to the difficulty of the original text, many students use "bridge" books before tackling Federer directly. If you find the PDF too dense, consider these resources: federer geometric measure theory pdf
E-book versions are available through major retailers for those who want a permanent, searchable digital copy. Modern Alternatives and Supplements
Federer introduced currents as generalized surfaces. Technically, they are continuous linear functionals on the space of differential forms. This allows mathematicians to use tools from functional analysis to solve geometric problems. A modern take that is highly recommended for
The notation is incredibly precise but can be overwhelming for beginners.
I can then recommend the specific chapters or supplementary notes that will be most helpful for your level! If you find the PDF too dense, consider
To help you get started with the right materials, I can provide more specific guidance if you tell me: Are you a or an undergraduate ?