Ch2

Question 1

Measurable Function

Answer 1

Let f:\Omega->\Omega’, and Omega and Omega’ have measurable spaces F, F’. Then f is measurable if f^-1(A)\in F for all A\inF’.
that is, the preimage of all sets in the measure space belong to the measure space of the domain.

Question 2

Alternative definition for Measurable Functions

Answer 2

If scriptA generates F’, then for all A\in scriptA, if f^-1(A)\inF, then we have a measurable function.
For borel sets, this is a convenient reduction since we only need to show it holds true for certain A, rather than all the elements in the borel set.

Question 3

Measurable functions include (6)

Answer 3

1) composition of measurable functions; 2) left and right, upper and lower semi-continuous functions in R; 3) monotone functions in R; 4) infimum and supremum; 5) point-wise limit (and hence stronger limit) for sequences of functions; 6) sums of sequences of functions that are positively valued.

Question 4

Sigma Algebras Generated by Measurable Maps

Answer 4

Let f:\Omega->\Omega’ be a measurable map. Then the set F={f^-1(A):A\inF’} is a sigma algebra within Omega.
Idea: it’s a sigma-algebra because by definition, any element that is contained within satisfies the complementarity and countable unions condition that is imposed on F.

Question 5

Measurable Function is integrable if

Answer 5

\int f+d\mu and \int f- d\mu are each integrable.

Question 6

Integral of a function

Answer 6

The supremum of functions in SF+(f) (all simple functions less than or equal to f) and their corresponding integral, which is defined as the finite dot product of the measures and set values.

Question 7

Simple functioons

Answer 7

Functions which are defined as the sum of indicator functions on finite partitions, each times some value a_i

Question 8

Monotone Convergence Theorem

Answer 8

Let there be a sequence of measurable functions which are increasing point wise to a limit f. These functions are positively valued. Then f is measurable and moreover the integral of f equals the limit of the integral of the functions in the sequence.

Question 9

Given any measure f:\Omega -> [0,\infty] there is a sequence of non-negative simple functions increasing pointwise to f. What’s the example? Why is this important?

Answer 9

f_n(w)=k2^-n if k2^-n \leq f(w) \leq (k+1)2^-n, else n.

Useful for discussing a.e. or approximating problems and applying MCT.

Question 10

Under what circumstances does linearity hold?

Answer 10

integrable functions mapping into R^*

Question 11

Fatou’s Lemma

Answer 11

Let {f_n} be a sequence of measurable functions into the positive reals. Then \int liminf f_n \leq liminf \int f_n

Question 12

Proof of Fatou’s Lemma

Answer 12

Use MCT and construct g_n = \inf f_n.

Note that g_n is increasing for all values of n. Hence int lim g_n \leq lim \int g_n.

Question 13

Dominated Convergence Theorem

Answer 13

Let {f_n} be a sequence of measurable functions which map to R*. Suppose they converge pointwise to a function f, and that there is an integrable function h that is bounded and |f|\leq h everywhere.
The \lim\int f_nd\mu = \int f d\mu and moreover the limit of the integral of their absolute difference is equal to 0.

Question 14

Almost everywhere

Answer 14

If the complement of the event has measure 0. Does not say that it is empty, remember: Lebesgue measure of countable number of points is zero, as we had countable additivity.

Question 15

When do two integrable functions have the same integral value?

Answer 15

when they are equal a.e.