# Category Archives: Measure Theory

## IMT Exercise 1.1.4

Let , and let , be elementary sets. Show that is elementary , and . Let partitions of and into finite sets of disjoint boxes be , and . Then Consider and , then It is clear that and as … Continue reading

## IMT Exercise 1.1.3

(Uniqueness of elementary measure). Let . Let be a map from the collection of elementary subsets of to the non-negative reals that obeys the non-negativity, finite additivity and translation invariance properties. Show that there exists a constant such that for … Continue reading

## IMT Exercise 1.1.2

Give an alternate proof of Lemma 1.1.2(ii) by showing that any two partitions of into boxes admit a mutual refinement into boxes that arise from taking Cartesian products from elements from finite collections of disjoint intervals. Lemma 1.1.2(ii) If is … Continue reading

## IMT Interval Length

In IMT, on page 5, it is stated that the length of any interval , is given by where and denotes the cardinality of a finite set . For smaller values of this can be directly tested using a spreadsheet. … Continue reading

## IMT Exercise 1.1.1

(Boolean closure). Show that if are elementary sets, then the union , the intersection , and the set theoretic difference and the symmetric difference are also elementary. If , show that the translate is also an elementary set. An elementary … Continue reading

## IMT Exercise 0.0.2

(Tonelli’s theorem for series over arbitrary sets). Let be sets (possibly infinite or uncountable), and be a doubly infinite sequence of extended non-negative reals indexed by and . Show that We know that Let with and finite and and . … Continue reading

## IMT Exercise 0.0.1

If is a collection of numbers such that , show that for all but at most countably many , even if itself is uncountable. and so none of the is equal to , i.e. . Let the sum . Define … Continue reading