- Show that ∫
. Hint: First verify the integral is finite. You
might use monotone convergence theorem to do this. It is easier than the stuff
you worried about in beginning calculus. Next let I = ∫
dx so that
I2 = ∫
dxdy. Now change the variables using polar coordinates.
It is all justified by the big change of variables theorem we have done. This becomes
an easy problem when you do this.
- Show that ∫
dx = 1. Here σ is a positive number called the
standard deviation and μ is a number called the mean. Hint: Just use the above
result to find ∫
dx and then change the variables in this one.
- The Gamma function is
Verify that this function is well defined in the sense that it is finite for all α > 0. Next
and Γ = Γ
x. Conclude that Γ =
n! if n is an integer. Now consider
Γ. This is
Change the variables in this integral. You might let t = u2. Then consider the above
- Let p,q > 0 and define B =
q−1. Show that
Hint: It is fairly routine if you start with the left side and proceed to change
- Let E be a Lebesgue measurable set in ℝ. Suppose m(E) > 0. Consider the
Show that E −E contains an interval. This is an amazing result. Recall the case of the
fat Cantor set which contained no intervals but had positive measure. Hint: Without
loss of generality, you can assume E is bounded. Let
Explain why f is continuous at 0 and f(0) > 0 and use continuity of translation in L1.
To see it is continuous, Now explain why this is small whenever
− x is small due to continuity of
translation in L1. Thus
> 0 and so by continuity, f > 0
near 0. If the integral is nonzero, what can you say about the integrand? You
must have for all x ∈ both
x + t ∈ E and t ∈ E. Now consider this a
- Does there exist a closed uncountable set which is contained in the set of irrational
numbers? If so, explain why and if not, explain why. Thus this uncountable set has no
rational number as a limit point.
- Find the area of the bounded region R, determined by 5x + y = 1,5x + y = 9,y = 2x,
and y = 5x.
- Here are three vectors.
T. These vectors determine a
parallelepiped, R, which is occupied by a solid having density ρ = y. Find the mass of
this solid. To find the mass of the solid, you integrate the density. Thus, if P is this
parallelepiped, the mass is
Hint: Let h :
3 → P be given by h =
then by definition of what is meant by a parallelepiped,
P and h is one to
one and onto.
- Find the volume of the ellipsoid +
. Hint: Change variables
to reduce to finding the volume of a sphere of radius 1 which you know is
- If f ≥ 0 is Lebesgue measurable, show there exists 0 ≤ g ≤ f such that g is Borel
measurable and g = f off a set of measure zero.
- Suppose f,g ∈ L1
. Define f ∗ g by
First show using the preceding problem that there is no loss of generality in assuming
that both f,g are Borel measurable. Next show this makes sense a.e. x and that in fact
for a.e. x
Hint: You can use Fubini’s theorem to write
- Suppose X : where
P is a probability measure and suppose X : Ω → ℝ is
measurable. that is, X−1
∈ℱ. Then consider the distribution measure λX
defined on the Borel sets of ℝp and given as follows. λX =
P. Explain why
this is a probability measure on
ℬ. Next show that if
X ∈ L1
Hint: Recall that from the definition of the integral,
Thus both ∫
ℝx+dλX = ∫
0∞xdλX and ∫
ℝx−dλX = ∫
dλX are finite. Now
from the definition of the integral again,
Now consider a similar thing for ∫
0∞xdλX and explain why
- Suppose you have a closed ball B centered at a point a and you have a continuous map
g : B
≡ B. Show that if
B ≡ maxx∈B is sufficiently
⊆ g if
δ < r. Hint: For x ∈B, estimate x − g +
≤ δ. Show it maps B to B. Then use the Brouwer fixed point theorem. It says
that if g doesn’t move any point very much, then g of the big ball contains a smaller
- Suppose you have two sets A,B in ℝn and A ⊆ B and there exists r : B → A such that
r is continuous and r leaves points of A unchanged. Suppose that B has the fixed point
property meaning that if f : B → B is continuous, then there exists a fixed point for f.
Show that then A also has the fixed point property. Use Problem 12 on Page 169
to verify that every closed, convex, bounded set in ℝn has the fixed point
property. Give other examples of such sets where B is a closed ball centered at 0.
- The next few problems involve invariance of domain. Suppose U is a nonempty open set
in ℝn,f : U → ℝn is continuous, and suppose that for each x ∈ U, there is a ball Bx
containing x such that f is one to one on Bx. That is, f is locally one to one. Show that
f is open.
- ↑ In the situation of the above problem, suppose f : ℝn → ℝn is locally one
to one. Also suppose that lim
∞. Show that it follows that
ℝn. That is, f is onto. Show that this would not be true if f is only
defined on a proper open set. Also show that this would not be true if the
∞ does not hold. Hint: You might show that f
is both open and closed and then use connectedness. To get an example in
the second case, you might think of
ex+iy. It does not include 0 + i0. Why
- ↑ Show that if f : ℝn → ℝn is C1 and if Df exists and is invertible for all
x ∈ ℝn,
then f is locally one to one. Thus, from the above problem, if lim
then f is also onto.
- In the proof of the invariance of domain principle, there was a function which was one to
one on a compact set. Then the assertion was made that its inverse was continuous.
Why is this so?
- Show that there is no one to one continuous function
such that f is onto.
- You know from linear algebra that there is no onto linear mapping A : ℝm → ℝn for
n > m. Show that there is no locally one to one continuous mapping which will map ℝm
- Suppose U is a nonempty set in ℝn. Let ∂U consist of the points p ∈ ℝn such that
B contains points of
U as well as points of ℝn ∖ U. Show that U is contained in
the union of the interior of U, denoted as int with
∂U. Now suppose that
f : U → ℝn and is one to one and continuous. Explain why int equals