Hello, I will (try to) keep a list of topics we have covered each week here:

Week 1:

We have seen that it is not possible to measure all subsets of R^n.

Recalled Riemann integration and shortcomings of the Riemann Integral.

Defined sigma algebras and measure spaces. Proved simple properties of measures.

Week 2

Defined the outer measure on R^n and explored its properties.

Week 3

Defined measurable subsets and the Lebesgue measure. Proved that measurable subsets differ from

open subsets by arbitrary small measure.

Week 4

Proved regularity properties of the Lebesgue measure. Introduced the Cantor set. Showed that the Lebesgue sigma algebra has cardinality equal to P(R^n). Introduced elementary families and pre-measures.