Exercise 3 Let (Ek) De a countable collection of measurable sets such that the measure m (UE) is finite. Define the sets F₁ = UE and G₁ = Ek. (1) (a) Show that, for all n € N, G₁ CE, CFCUE. (b) Show that (F) is descending and (G₁), is ascending. (2) Determine (3) Assume that lim sup E, lim inf E, = E. Show that 8-90 Reminder: ™ (UC) and (².). mUG₁ m lim sup E 11-400 m(E)= lim m(En). - -ñ (Ù₂) n=1 and lim inf E -Ů (ñ). n=1
Exercise 3 Let (Ek) De a countable collection of measurable sets such that the measure m (UE) is finite. Define the sets F₁ = UE and G₁ = Ek. (1) (a) Show that, for all n € N, G₁ CE, CFCUE. (b) Show that (F) is descending and (G₁), is ascending. (2) Determine (3) Assume that lim sup E, lim inf E, = E. Show that 8-90 Reminder: ™ (UC) and (².). mUG₁ m lim sup E 11-400 m(E)= lim m(En). - -ñ (Ù₂) n=1 and lim inf E -Ů (ñ). n=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Exercise 3
Let (Ek) De a countable collection of measurable sets such that the measure
m (UE) is finite. Define the sets
(1) (a) Show that, for all n € N,
(b)
(2)
(3)
F₁ = UE and G₁ = n Ek.
kn
Show that (F), is descending and (Gn), is ascending.
Determine
Reminder:
G, CƠN CẢ, CŨE
k=1
lim sup E
11-400
Assume that lim sup En =lim inf E, E. Show that
=
11-98
m (ŨG) and (³.).
m
(₁).
m(E)= lim m(En).
71-80
=
-ñ (U₂)
n=1
and lim inf E,
11-48
=
-Ů (ñ).
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