Introduction to Analysis and Probability - Notation and Terminology - Mathematics Assignment Help

Asssignment Task:

Task:

Measurable Sets

At any given time point, we are interested in knowing what the future will be at a later time point. But the future is full of uncertainties: tomorrow it may rain or may not rain; the lottery ticket you are buying now may win or may not win. Just like flipping a coin, we will not know which scenario would eventually become true, until the coin has been flipped. Thus, in the presence of uncertainties, instead of asking what the future will be, we shall ask what are the possible scenarios and what are their chances to become true in the future.

1. Definition and basic properties

Suppose that we are going to conduct an experiment of flipping a coin three times. We use H and T to denote head and tail, respectively. Then we are facing eight possible outcomes in the future: HHH, HHT, HTH,HTT, THH,THT,TTH,TTT. We collect them together and denote it by a set
Ω := HHH, HHT, HT H, HT T, T HH, T HT, T TH, T T T
(1.1) .
Consider the event that the first flip is H. It means precisely that if we have conducted the experiment, then our final outcome would be one of the following four: HHH, HHT, HTH,HTT. We may thus use the set

FH := HHH, HHT, HT H, HT T

to denote the event that the first flip is H. Other events can be similarly identified as subsets of Ω as well. For example, we identify the event that the second flip is H with the set SH := {HHT, HHH, T HH, T HT}.

Monotone class theorem

It is generally extremely difficult to figure out all the elements in a generated σ-algebra. A general approach to get around this difficulty is to study another collection of subsets of Ω that contains the generators of the σ-algebra in question but satisfies some other properties and then compare this collection with the generated σ-algebra. We present a monotone class theorem in this spirit. It will be used later a few times.

Exercises

1.1. Let F be a σ-algebra over a set Ω and E be a non-empty set in F.
Then F|E := {F : F ∈ F, F ⊂ E} is a σ-algebra over E. Note that the universal set for F|E is E, not Ω.

1.2. Let Ω be as in (1.1). Let TH be the event that the third flip is H. Show that P(Ω) = σ({FH, SH, TH}).

1.3. Let Ω be a non-empty set. Let {An}n∈N be a given partition of Ω,
i.e., Ω = ∪n∈NAn and Aj ∩ Ak = ∅ for any distinct j, k in N. Show that

σ {An}n∈N =n [j∈JAj : J ⊂ No
.

1.4. Show that
B =σ({(a, ∞) : a ∈ R}) = σ({[a, ∞) : a ∈ R})=σ({(−∞, a) : a ∈ R}) = σ({[a, b) : a, b ∈ R, a < b}).

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