Real Analysis Part 5 (Lebesgue Measure)

Define the Lebesgue outer measure and measurable sets. Show E is measurable iff for A ⊆ R, m*(A) = m*(A∩E) + m*(A∩E^c), and that empty and universal sets are measurable.

Show that a set with zero outer measure is measurable by applying outer measure properties, De Morgan's law, and countable subadditivity in Lebesgue measure.

Prove that if E1 and E2 are measurable, their union E1 ∪ E2 is measurable by using the outer measure definition, De Morgan's law, and subadditivity.

Show that the family m of measurable sets forms an algebra of sets by proving complements and unions of measurable sets remain measurable.

Establishes that a finite sequence of pairwise disjoint measurable sets yields that outer measure of a intersected with union equals sum of outer measures of a intersected with each set.

Show that the collection of measurable sets forms a sigma algebra or Borel field by proving closure under complements and countable unions, using outer measure to establish measurability.

Proves the interval (a, infinity) is measurable by showing outer measure of A equals the sum of A ∩ (a, infinity) and A ∩ (a, infinity)^c, via A1 and A2.

Demonstrate that every open set is measurable and that all Borel sets are measurable by showing open intervals form a sigma algebra and using complements, unions, and intersections.

Demonstrate that for a sequence of measurable sets, the Lebesgue measure of the union is at most the sum of their measures, with equality for pairwise disjoint sets.

Real analysis part 5 shows that a decreasing sequence of measurable sets has the Lebesgue measure of the intersection equal to the limit of the individual measures, via disjoint differences.

Proves the Lebesgue measure relation for measurable sets E1 and E2: m(E1 ∪ E2) + m(E1 ∩ E2) = m(E1) + m(E2) via subset and disjointness arguments.

Show that disjoint measurable sets E_i yield outer measure of A ∩ ⋃ E_i equals the sum of outer measures of A ∩ E_i, using finite case and countable subadditivity.

Define sum modulo one for numbers in [0,1) and translate a measurable subset E by y to E+y, proving Lebesgue measure is translation invariant.

Show the existence of a non-measurable set under a translation-invariant, countably additive Lebesgue measure on a sigma-algebra; construct a representative set from equivalence classes on [0,1] and derive a contradiction.

Learn how extended real valued functions on a measurable domain are Lebesgue measurable, proving the equivalence of sets {f>α}, {f≥α}, {f<α}, {f≤α} for all α.

Show that a continuous function on [0,1] with a measurable domain is measurable by proving {x: f(x) > alpha} is open, hence a Borel, hence measurable.

Learn that the sum, difference, product, and scalar multiples of two real-valued measurable functions are measurable on the same domain, and how their domains align under addition, subtraction, and multiplication.

For a sequence of measurable functions with the same domain, prove the measurability of the pointwise supremum and infimum. Then show the limsup and liminf of the sequence are measurable.

Demonstrate that if a measurable function f equals g almost everywhere under Lebesgue measure, then g is measurable by using the measure-zero set where they differ.

Explore Littlewood's three principles for Lebesgue measure, including near intervals, near continuity, and near uniform convergence; learn a key almost everywhere convergence result on finite measure sets.