User:MCPOliseno /Math710 Essay

Post Midterm Essay Questions

1 Measure:

(a) Summarize the notion of Lebesgue outer measure and the role of Caratheodoryís measurability criterion. You should explain what is gained by restriction of outer measure to the class of measurable sets.

Lebesgue outer measure: m*(A) = inf {$ \sum \ $ t $ (A_n) \ $}, where the infimum is taken over all countable collections of open intervals $ (A_n) \ $, such that $ A \subset \cup A_n \ $ and $ tA_n \ $ is the standard length of the interval $ A_n \ $. Lebesgue outer measure is countably subadditive rather than finitely subaditive, like Jordan outer measure. Since outer measures are countably subadditive, rather than countably additive, they are weaker than measures, but outer measure can measure all subsets of a set, unlike measure which can only measure a $ \sigma \ $-algebra of measurable sets. Caratheodoryís measurability criterion is shown by, let $ \mu \ $* be an outer measure on a set X. A set E $ \subset \ $ X is said to be Caratheodoryís measurable with respect to $ \mu \ $* if one has $ \mu \ $*(A) = $ \mu \ $*(A $ \cap \ $ E) + $ \mu \ $* (A\E), for every set A $ \subset \ $ X.

(b) Explain why it is desirable that class $ M \ $ of measurable sets forms a $ \sigma \ $-algebra (Hint: given a sequence {$ f_n \ $} of, say, continuous functions, think about what is involved in finding m ({x:$ \alpha \ $ < lim inf $ f_n \ $ (x) < $ \beta \ $})

Because when taking compliments of countable unions and countable intersections, you need the $ \sigma \ $-algebra to keep it closed, since the compliment of countable unions is not necessarily a countable union, like for example the Cantor set.

(c) Why is desirable that $ M \ $ contains the Borel sets?

The Borel set is the the smallest $ \sigma \ $-algebra that contains all open sets, it also contains the smallest $ \sigma \ $-algebra that contains all the closed sets and the smallest $ \sigma \ $-algebra that contains the open intervals. Since $ M \ $ is a $ \sigma \ $-algebra, then each Borel set must be contained in $ M \ $.

2 The Lebesgue integral:

(a) Following the presentation given in class, compare and contrast the definitions of Lebesgue and Riemann integrals for bounded functions on an interval [a, b].

Lebesgue is done with simple functions, not step functions, like the Riemann integral. A simple function is a valued function over a subset of the real line, which attains only a finite number of values. A step function is a piecewise constant function having only finitely many pieces. Simple functions are always measurable. The simple functions are used in integration (Lebesgue Integrals), since it is easy to create a definition of an integral for simple functions and it is also easier to generate more general functions by using sequences of simple functions. The relation of the simple function to the Lebesgue Integral is that when dealing with a non-negative measurable function, $ f: X \to \R \ $, is the pointwise limit of a monotone increasing sequence of simple functions.

Note that all step functions are simple and thus Riemann is a special case of the Lebesgue Integral. Also note that Riemann partitions the domain of the integral, unlike the Lebesgue integral that partitions the range.

The Riemann integral can fail to meet our needs. Firstly, the class of Riemann integrable functions is relatively small. Second and related to the first, the Riemann integral does not have satisfactory limit properties. That is, given a sequence of Riemann integrable functions {$ f_n \ $} with a limit function f = $ lim_{n \to \infty} f_n \ $ it does not necessarily follow that the limit function f is Riemann integrable. The Lebesgue integral solves many of the problems left by the Riemann integral.

(b) Show that upper and lower Lebesgue integrals defined on page 79 of the text are equivalent to the definition given in class via measurable partitions.

Let $ P \ $ = {$ x_0, x_1, \dots, x_n \ $} be a partition of [a, b].

The upper sum, $ \overline{S} \ $ (f, P) = $ \sum_{i=1}^{n} M_i \ $ (f)($x_i - x_{i-1} \ $), where $ M_i \ $ = sup f

and the lower sum, $ \underline{S} \ $ (f, P) = $ \sum_{i=1}^{n} m_i \ $ (f)(($x_i - x_{i-1} \ $), where $ m_i \ $ = inf f

$ \int_{a}^{\overline{b}} \ $ f = inf $ \overline{S} \ $ (f, P) and $ \int_{\underline{a}}^{b} \ $ f = $ \underline{S} \ $ (f, P).

if $ \int_{a}^{\overline{b}} \ $ f = $ \int_{\underline{a}}^{b} \ $ f, then $ f \ $ is integrable $ \iff \ $ if $ \forall \epsilon \ $ > 0, $ \exists P \ $ such that $ \overline{S} \ $ (f, P) - $ \underline{S} \ $ (f, P) < $ \epsilon \ $.

Upper and Lower Integrals are defined in the book as follows:

Let $ f: E \to \R \ $ be a bounded real-valued function and $ E \ $ is a measurable set of finite measure. $ \psi \ $ and $ \varphi \ $ are simple functions.

Upper Integral: inf $ \int_{E} \psi \ $, where $ \psi \ge f \ $

Lower Integral: sup $ \int_{E} \varphi \ $, where $ \varphi \ge f \ $.

If the function is Integrable, then the Upper Integral = Lower Integral, which is exactly what the definition presented in class proves.

(c) One might define a simple function without the restriction of measurability then define $ \int \ $ $ \sum \ $ $ c_i \ $ $ X_{A_i} \ $ : = $ \sum c_i \ $ m*($ A_i \ $). Which standard properties of the integral might go wrong?

3 Extensions: Let $ A \ $ be a $ \sigma \ $-algebra of subsets of set $ X \ $ and $ \mu : A \to \ $ [0, $ \infty \ $]. We say $ \mu \ $ is the measure if $ \mu \ $ is countably additive (for disjoint sequences of sets in $ A \ $) and $ \mu (\varnothing) \ $ = 0.

(a) Show that $ \mu \ $ satisfies the "continuity of measure" property.

(b) In context of $ A \ $ and $ \mu \ $ what conditions should a real-valued function on $ X \ $ satisfy in order to be classified as measurable?

(c) Mimicing the development of the Lebesgue integral, outline how would you define $ \int_x \ $ $ f d\mu \ $ for a measurable real-valued function on $ X \ $ ?