A fancy name for calculus, with the "more advanced" connotation.
A ring where multiplication is commutative and there is always an inverse.
A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.
And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.
Basically the nicest, least restrictive, 2-operation type of algebra.
Examples:
Main motivation: Lebesgue integral.
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The key idea, is that we can't define a measure for the power set of R. Rather, we must select a large measurable subset, and the Borel sigma algebra is a good choice that matches intuitions.
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