STA 7466: Probability Theory 1
Assignment 1
Problems to Turn In
Due 2023-09-18 (Monday).
1. Let 𝑓(A)represent the field generated by a class Ain 𝛺. Show that for nonempty A,
𝑓(A)is the class of sets of the form Ò𝑚
𝑖=1Ñ𝑛𝑖
𝑗=1𝐴𝑖 𝑗 , where for each 𝑖and 𝑗either 𝐴𝑖 𝑗 ∈Aor
𝐴𝑐
𝑖 𝑗 ∈A, and where the 𝑚sets Ñ𝑛𝑖
𝑗=1𝐴𝑖 𝑗 ,1≤𝑖≤𝑚, are disjoint. The sets in 𝑓(A)can thus
be explicitly presented, which is not in general true of the sets in 𝜎(A).
Hint: Let Cbe the class of all sets of the specified form. Then A⊂C(for any 𝐴∈C, take
𝑚=1,𝑛1=1, and 𝐴1,1=𝐴), and because 𝑓(C)must be closed under the formation of finite
unions and intersections, it is clear that A⊂C⊂𝑓(A). Thus, it suffices to show that Cis
a field. Because Ais nonempty, there exists at least one set 𝐴∈Aand thus 𝛺=𝐴⊎𝐴𝑐∈C.
Next show that that Cis closed under finite intersections and then use this fact to show that
Cis closed under complementation.
2. Show that if 𝐵∈𝜎(A), then there exists a countable subclass A𝐵of Asuch that 𝐵∈𝜎(A𝐵).
Hint: Use a good-sets argument.
3. (a) Let Fbe the field consisting of the finite and the co-finite sets in an infinite 𝛺, and
define 𝑃on Fby taking 𝑃(𝐴)to be 0or 1as 𝐴is finite or cofinite. (Note that 𝑃is not
well defined if 𝛺is finite, since then every subset of 𝛺is both finite and cofinite.) Show
that 𝑃is well definied and finitely additive.
(b) Show that this 𝑃is not countably additive if 𝛺is countably infinite.
(c) Show that this 𝑃is countably additive if 𝛺is uncountable.
(d) Now let Fbe the 𝜎-field consisting of the countable and the cocountable sets in an
uncountable 𝛺, and define 𝑃on Fby taking 𝑃(𝐴)to be 0or 1as 𝐴is countable or
cocountable. (Note that 𝑃is not well defined if 𝛺is countable, since then every subset
of 𝛺is both countable and co-countable.) Show that 𝑃is well defined and countably
additive.
4. Prove Proposition 1.3.4 in the course notes. In other words, prove that Lsatisfies the new
𝜆-system postulates:
(𝜆1)𝛺∈L;
(𝜆2)𝐴∈Limplies 𝐴𝑐∈L;
(𝜆3)𝐴1, 𝐴2,· · · ∈ Land 𝐴𝑚∩𝐴𝑛=∅for all 𝑚≠𝑛, imply Ò𝑛𝐴𝑛∈L.
if and only if it satisfies the old 𝜆-system postulates:
(𝜆′
1)𝛺∈L;
(𝜆′
2)if 𝐴, 𝐵 ∈Land 𝐴⊂𝐵, then 𝐵−𝐴∈L;
(𝜆′
3)if 𝐴1, 𝐴2,· · · ∈ Land 𝐴𝑛↑𝐴, then 𝐴∈L.
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