- Show that ∫
_{0}^{∞}e^{−x2 }dx =. 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 = ∫_{0}^{∞}e^{−x2 }dx so that I^{2}= ∫_{0}^{∞}∫_{0}^{∞}e^{−(x2+y2) }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 ∫
_{−∞}^{∞}e^{− 2 (x−2μσ)2- }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 ∫_{−∞}^{∞}e^{−x2 }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 verify that

and Γ

= Γx. Conclude that Γ= n! if n is an integer. Now consider Γ. This isChange the variables in this integral. You might let t = u

^{2}. Then consider the above problem. - Let p,q > 0 and define B= ∫
_{0}^{1}x^{p−1}^{q−1}. Show thatHint: It is fairly routine if you start with the left side and proceed to change variables.

- Let E be a Lebesgue measurable set in ℝ. Suppose m(E) > 0. Consider the
set
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 L

^{1}. To see it is continuous,− x is small due to continuity of translation in L^{1}. Thus f= m> 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 little. - 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},^{T}, and^{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 isHint: Let h :

^{3}→ P be given by h= t_{1}+ t_{2}+ t_{3}then by definition of what is meant by a parallelepiped, h= P and h is one to one and onto. - Find the volume of the ellipsoid ++= 1 . Hint: Change variables to reduce to finding the volume of a sphere of radius 1 which you know isπ.
- 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 ∈ L
^{1}. Define f ∗ gbyFirst 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

Next show

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 λ_{X}defined on the Borel sets of ℝ^{p}and given as follows. λ_{X}= P. Explain why this is a probability measure on ℬ. Next show that if X ∈ L^{1},Hint: Recall that from the definition of the integral,

Thus both ∫

_{ℝ}x^{+}dλ_{X}= ∫_{0}^{∞}xdλ_{X}and ∫_{ℝ}x^{−}dλ_{X}= ∫_{−∞}^{0}dλ_{X}are finite. Now from the definition of the integral again,Now consider a similar thing for ∫

_{0}^{∞}xdλ_{X}and explain why - Suppose you have a closed ball B centered at a point a and you have a continuous map
g : B→ B≡ B. Show that if
_{B}≡ max_{x∈B}is sufficiently small, then B⊆ gif δ < r. Hint: For x ∈, estimate x − g+ b where≤ δ. Show it maps to . 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 ball. - Suppose you have two sets A,B in ℝ
^{n}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 ℝ^{n}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 ℝ
^{n},f : U → ℝ^{n}is continuous, and suppose that for each x ∈ U, there is a ball B_{x}containing x such that f is one to one on B_{x}. That is, f is locally one to one. Show that fis open. - ↑ In the situation of the above problem, suppose f : ℝ
^{n}→ ℝ^{n}is locally one to one. Also suppose that lim_{|x| →∞}= ∞. Show that it follows that f= ℝ^{n}. 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 condition lim_{|x| →∞}= ∞ does not hold. Hint: You might show that fis both open and closed and then use connectedness. To get an example in the second case, you might think of e^{x+iy}. It does not include 0 + i0. Why not? - ↑ Show that if f : ℝ
^{n}→ ℝ^{n}is C^{1}and if Dfexists and is invertible for all x ∈ ℝ^{n}, then f is locally one to one. Thus, from the above problem, if lim_{|x| →∞}= ∞, 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 : ℝ
^{m}→ ℝ^{n}for n > m. Show that there is no locally one to one continuous mapping which will map ℝ^{m}onto ℝ^{n}. - Suppose U is a nonempty set in ℝ
^{n}. Let ∂U consist of the points p ∈ ℝ^{n}such that Bcontains points of U as well as points of ℝ^{n}∖ U. Show that U is contained in the union of the interior of U, denoted as intwith ∂U. Now suppose that f : U → ℝ^{n}and is one to one and continuous. Explain why intequals f.

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