Probability Theory and Stochastic Processes
LIST 2
(1) Show that any simple function is measurable.
(2) Let (Ω, F, µ) be a measure space where µ is the counting measure. Consider A = {a1 , a2 , a3 } ⊂ Ω and a measurable function
f.
(a) Is XA f a simple function?
R
(b) Compute A f dµ.
(3) Let (Ω, F, µ) be a measure space where µ is the Dirac measure
δ on a ∈ Ω. Compute
R
(a) ϕ dµ where ϕ is a simple function.
R
(b) f dµ where f : Ω → R is any function.
(4) (Markov inequality) Let (Ω, F, µ) be a measure space and a
measurable function f : Ω → R such that f ≥ 0. Show that for
any λ > 0 we have
Z
1
f dµ.
µ ({x ∈ Ω : f (x) ≥ λ}) ≤
λ
(5) Show that a function f : Ω → R is measurable iff f + and f − are
measurable. (Hint: f + = XA f where A = {x ∈ Ω : f (x) > 0})
(6) Let f be a measurable function. Show that
(a) |f | is also measurable.
(b) f −1 ({a}) = {x : f (x) = a} is a measurable set.
(7) Let A ⊂ F be σ-algebras.
(a) Show that if a function is A-measurable, then it is also
F-measurable.
(b) Is the converse true? If not, write an example of a function
which is F-measurable but not A-measurable.
(8) Let (Ω, F, µ) be a measure space, A ⊂ F a σ-subalgebra, f, g
are F-measurable functions and h is A-measurable. Show that
R
R
(a) If B f dµ = B g dµ for every B ∈ F, then f = g a.e.
R
R
(b) If A f dµ = A h dµ for every A ∈ A, then f might not be
equal to h a.e. Find an example.
1
2
(9) Use the dominated convergence theorem to determine the limits
(where δA stands for the Dirac delta measure on A):
R +∞ n
(a) lim 0 1+rr n+2 dr
n→+∞
R +∞ n
(b) lim 0 1+rr n+2 dδ{1} (r)
n→+∞
Rπ √
n
(c) lim 0 1+xx2 dx
n→+∞
Rπ √
n
(d) lim 0 1+xx2 dδ{0} (x)
n→+∞
R +∞
(e) lim −∞ e−|x| cosn (x) dx
n→+∞
R +∞
(f) lim −∞ e−|x| cosn (x) dδ{0} (x)
n→+∞
R
2
2 n
(g) lim R2 e−(x +y ) dδC (x, y) where C is the unit circle cenn→+∞
tered at the origin.
R
n (x−y)
(h) lim R2 1+cos
2 +y 2 +1)2 dδA (x, y) where A is the straight line
(x
n→+∞
y = x.