Econ 508B: Lecture 2 Lebesgue Measure and Lebesgue Measurable Functions

Hongyi Liu

Washington University in St. Louis

July 18, 2017

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 1 / 35 Outline

1 Review of Real Analysis

2 Lebesgue Outer Measure

3 Lebesgue Measure

4 Lebesgue Measurable Functions

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 2 / 35 Outline

1 Review of Real Analysis

2 Lebesgue Outer Measure

3 Lebesgue Measure

4 Lebesgue Measurable Functions

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 3 / 35 Remark 1.1 The elements of C are called open sets and the collection C is called a topology on S. Using de Morgan’s laws, the above axioms defining open sets become axioms defining closed sets.

Topological Space

Definition 1.1 A topological space is a pair (S, C), where S is a non-empty set and C is a collection of subsets of S such that ∅, S ∈ C, (finite intersection:)C1,C2 ∈ C ⇒ C1 ∩ C2 ∈ C, and + (finite or infinite union:){Ck : k ∈ N } ⊂⇒ ∪k∈N + ∈ C

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 4 / 35 Topological Space

Definition 1.1 A topological space is a pair (S, C), where S is a non-empty set and C is a collection of subsets of S such that ∅, S ∈ C, (finite intersection:)C1,C2 ∈ C ⇒ C1 ∩ C2 ∈ C, and + (finite or infinite union:){Ck : k ∈ N } ⊂⇒ ∪k∈N + ∈ C

Remark 1.1 The elements of C are called open sets and the collection C is called a topology on S. Using de Morgan’s laws, the above axioms defining open sets become axioms defining closed sets.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 4 / 35 Union/Intersection

Definition 1.2

A sequence {Ck} of sets is said to increase to ∪kCk if Ck ⊂ Ck+1 for all k and to decrease to ∩kCk if Ck ⊃ Ck+1 for all k; we use the notations Ck % ∪kCk and Ck & ∩kCk to denote these two possibilities. ∞ If {Ck}k=1 is a sequence of sets, we define

∞  ∞  ∞  ∞  \  [  [  \  lim supC =  C  , lim infC =  C  k  k k  k j=1 k=j  j=1 k=j 

noting that S∞ T∞ Uj = k=j Ck and Vj = k=j Ck satisfy Uj & lim supCk and Vj % lim infCk,

lim infCk ⊂ lim supCk.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 5 / 35 Norm on RN

(1) |x| ≥ 0 and |x| = 0 ⇔ x = 0, N (2) |αx| = |α||x|, x ∈ R , α ∈ R, (3) |x + y| ≤ |x| + |y|, ∀x, y ∈ R, (4)( Cauchy-Schwarz inequality:)|x· y| ≤ |x||y|. Proof for (4): 1 1 ∀x, y ∈ , xy ≤ x2 + y2 R 2 2 X 1 X 1 X 1 1 x · y = x y ≤ x2 + y2 = |x|2 + |y|2 k k 2 k 2 k 2 2 1 x0 = λx, y0 = y, λ 6= 0(to be chosen), x0 · y0 = x · y λ s 1 1 |y| ≤ |λ|2|x|2 + |y|2 = |x||y| (choose λ = ) 2 2|λ|2 |x|

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 6 / 35 Metric Space

Definition 1.3 A metric space is a pair (S, d) where S is a nonempty set and d is a + function from S × S to R (d is called a metric on S) satisfying (i) d(x, y) = d(y, x) for all x, y ∈ S, (ii) d(x, y) = 0 iff x = y, (iii)( triangle inequality:)d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ S.

A metric space (S, d) is a topological space where a set C is open if for all x ∈ C, ∃ an  > 0 such that B(x, ) ≡ {y : d(y, x) < } ⊂ C.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 7 / 35 continued

Any one of the following metrics defined on any Euclidean space n R , 1 ≤ n < ∞ is a metric space: (1) For 1 < p < ∞,

1  n  p X p dp(x, y) =  |xi − yi|    i=1 (2) d∞(x, y) = max |xi − yi| 1≤i≤n (3) For 0 < p < 1,  n  X p dp(x, y) =  |xi − yi|    i=1 1 Question: why there does not exist power p to (3)?

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 8 / 35 Outline

1 Review of Real Analysis

2 Lebesgue Outer Measure

3 Lebesgue Measure

4 Lebesgue Measurable Functions

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 9 / 35 Lebesgue Outer (Exterior) Measure

Define closed n-dimensional intervals I = {x : aj ≤ xj ≤ bj, j = 1, ..., n} Qn and their volumes v(I) = j=1(bj − aj). To define the outer measure n of an arbitrary subset C of R , cover C by a countable collection S of intervals Ik, and let X σ(S) = v(Ik)

Ik∈S The Lebesgue outer measure of C, denoted as µ∗(C), is defined by

µ∗(C) = infσ(S)

where the infimum is taken over all covers S of C. Thus, 0 ≤ µ∗(C) ≤ +∞

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 10 / 35 Properties of Outer Measure

For an interval I, µ∗(I) = v(I). ∗ ∗ Monotonicity: If C1 ⊂ C2, then µ (C1) ≤ µ (C2).

Countable sub-additivity: If C = ∪Ck is a countable union of ∗ P ∗ sets, then µ (C) ≤ µ (Ck). Empty set: The empty set has outer measure zero, e.g., Q.

Remark 2.1 In particular, any subset of a set with outer measure zero has outer measure zero and that the countable union of sets with outer measure zero has outer measure zero as shown by the example of Q. Moreover, there are sets with outer measure zero that are not countable, e.g., Cantor Set.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 11 / 35 continued

Theorem 2.1 (Outer Approximation)

n Let C ⊂ R . Then given  > 0, ∃ an open set G s.t. C ⊂ G and µ∗(G) ≤ µ∗(C) + . Hence,

µ∗(C) = infµ∗(G),

where the infimum is taken over all open sets G containing C.

Theorem 2.2 n ∗ ∗ If C ⊂ R , ∃ a set H of type Gδ s.t. C ⊂ H and µ (C) = µ (H)

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 12 / 35 Outline

1 Review of Real Analysis

2 Lebesgue Outer Measure

3 Lebesgue Measure

4 Lebesgue Measurable Functions

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 13 / 35 Definition 3.2 (Measure) If C is measurable, its outer measure is referred to as its Lebesgue measure or simply its measure, and denoted by µ(C) as previously illustrated: µ(C) = µ∗(C), for measurable C.

Example 3.1 Every open set is measurable. Every set of outer measure zero is measurable.

‘Measurable’ & ‘Measure’

Definition 3.1 (Lebesgue measurable) n A subset C of R is defined to be Lebesgue measurable, or measurable, if given  > 0, ∃ an open set G s.t.

C ⊂ G, and µ∗(G − C) < 

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 14 / 35 Example 3.1 Every open set is measurable. Every set of outer measure zero is measurable.

‘Measurable’ & ‘Measure’

Definition 3.1 (Lebesgue measurable) n A subset C of R is defined to be Lebesgue measurable, or measurable, if given  > 0, ∃ an open set G s.t.

C ⊂ G, and µ∗(G − C) < 

Definition 3.2 (Measure) If C is measurable, its outer measure is referred to as its Lebesgue measure or simply its measure, and denoted by µ(C) as previously illustrated: µ(C) = µ∗(C), for measurable C.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 14 / 35 ‘Measurable’ & ‘Measure’

Definition 3.1 (Lebesgue measurable) n A subset C of R is defined to be Lebesgue measurable, or measurable, if given  > 0, ∃ an open set G s.t.

C ⊂ G, and µ∗(G − C) < 

Definition 3.2 (Measure) If C is measurable, its outer measure is referred to as its Lebesgue measure or simply its measure, and denoted by µ(C) as previously illustrated: µ(C) = µ∗(C), for measurable C.

Example 3.1 Every open set is measurable. Every set of outer measure zero is measurable.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 14 / 35 Remark 3.1 Noting that the condition for measurability should not be confused with theorem 2.1, which only states that ∃ an open set G containing C such that µ∗(G) ≤ µ∗(C) + . Since G = C ∪ (G − C) when C ⊂ G, which only implies that µ∗(G) ≤ µ∗(C) + µ∗(G − C). However, we cannot obtain from µ∗(G) ≤ µ∗(C) +  that µ∗(G − C) < .

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 15 / 35 Properties of Measurable Set

(Countable subaddtivity:) The union C = ∪Ck of a countable P measurable sets is measurable and µ(C) ≤ µ(Ck). Every closed set F is measurable. The complement of a measurable set is measurable.

The intersection C = ∩kCk of a countable measurable sets is measurable.

If C1 and C2 are measurable, then C1 − C2 is measurable. n The collection of measurable subsets of R is σ-algebra. Every Borel set is measurable.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 16 / 35 Properties of Lebesgue Measure

Lemma 3.1 n Set C in R is measurable if.f. given  > 0, ∃ a closed set F ⊂ C such that µ∗(C − F ) < .

Theorem 3.1

If {Ck} is a countable collection of disjoint measurable sets, then S P µ( k Ck) = k µ(Ck).

Corollary 3.1

Suppose C1 and C2 are measurable, C2 ⊂ C1, and µ(C2) < +∞. Then µ(C1 − C2) = µ(C1) − µ(C2).

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 17 / 35 continued

Theorem 3.2 ∞ Let {Ck}k=1 be a sequence of measurable sets.

(1) If Ck % C, then limk→∞ µ(Ck) = µ(C).

(2) If Ck & C and µ(Ck) < +∞ for some k, limk→∞ µ(Ck) = µ(C).

Proof: (1) Assume that µ(Ck) < +∞ for all k, otherwise both limk→∞ µ(Ck) and µ(C) are infinite and the statement holds. Break C via C = C1 ∪ (C2 − C1) ∪ ... ∪ (Ck − Ck−1) ∪ ... By theorem 3.1,

µ(C) = µ(C1) + µ(C2 − C1) + ... + µ(Ck − Ck−1) + ···. By corollary 3.1,

µ(C) = µ(C1)+(µ(C2)−µ(C1))+···+(µ(Ck)−µ(Ck−1))+··· = lim µ(Ck). k→∞

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 18 / 35 Proof: (2) clearly assume that µ(C1) < +∞. Write

C1 = C ∪ (C1 − C2) ∪ · · · ∪ (Ck − Ck+1) ∪ · · ·.

Likewise,

µ(C1) = µ(C) + (µ(C1) − µ(C2)) + ··· + (µ(Ck) − µ(Ck+1)) + ···

= µ(C) + µ(C1) − lim µ(Ck). k→∞

Hence, µ(C) = limk→∞ µ(Ck).

Noting that the condition µ(Ck) < +∞ for some k is necessary.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 19 / 35 Characterizations of Measurability

Theorem 3.3

C is measurable if and only if C = H − Z, where H is of type Gδ and µ(Z) = 0.

C is measurable if and only if C = H ∪ Z, where H is of type Fδ and µ(Z) = 0.

Proof: (sufficiency) It is trivial that C is measurable because H and Z are both measurable sets. (necessity): For the first one, suppose that C is measurable and choose an open set Gk such that C ⊂ Gk and µ(Gk − C) < 1/k for T k = 1, 2, .... Let H = k Gk, which is a set of type Gδ,C ⊂ H and H − C ⊂ Gk − C for every k. As k → ∞, µ(H − C) = 0. Secondly, C is measurable, so is the complement set of C, denoted as Cc. Then applying the result of first one, we obtain that c T S c C = k Gk − Z, and using de Morgan’s laws we have C = ( k Gk) ∪ Z.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 20 / 35 Nonmeasurable set

1 Construct a nonmeasurable subset of R , and the construction in d R , d > 1 is similar. The construction of a nonmeasurable set uses the following axiom of choice and rests on equivalence relation among real numbers in [0, 1]. Axiom 3.1 (Zermelo’s Axiom:) A family of arbitrary nonempty disjoint sets indexed by a set A, {Cα : α ∈ A}, ∃ a set consisting of exactly one element from each Cα, α ∈ A.

Lemma 3.2 1 Let C be a measurable subset of R with µ(C) > 0. Then the set of differences {d : d = x − y, x ∈ C, y ∈ C} contains an interval centered at the origin.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 21 / 35 Vitali Theorem: There exist nonmeasurable sets.

Proof: An equivalence relation, defined as x ∼ y on the real line if x − y is rational, which can be formulated by Cα = {α + r : r is rational}. It means that any two classes Cα and Cβ are either identical or disjoint.

Hence, one equivalence class consists of all the rational numbers, and the other distinct classes are disjoint sets of irrational numbers.

Using Zemelo’s axiom, construct the set C consisting of exactly one element from each distinct equivalence class, therefore, any two points of C must differ by an irrational number, which implies that the set {d : d = x − y, x ∈ C, y ∈ C} cannot contain an interval. According to Lemma 3.2 , it suffices that either C is not measurable or µ(C) = 0.

Let Cr = C + r, r ∈ Q, then ∪r∈QC representing the union of the 1 1 translation of C by every rational is R , R would have measure zero if C. Then it completes the proof that C is nonmeasurable.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 22 / 35 Outline

1 Review of Real Analysis

2 Lebesgue Outer Measure

3 Lebesgue Measure

4 Lebesgue Measurable Functions

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 23 / 35 Measurable Functions

Definition 4.1 (Measurable function)

In general, let Ω1 be a set with a σ-algebra C1, and Ω2 be a set with a sigma-algebra C2, and T be a function from Ω1 to Ω2. Say T is hC1, C2i−measurable if the inverse image {x ∈ Ω1 : T x ∈ C2} ∈ C1 for each C2 ∈ C2.

In particular, if Ω2 = R, C2 becomes the Borel σ-algebra B(R). Example 4.1 Recall the definition for random variable in the previous slide.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 24 / 35 Definition 4.2 n Let C be a measurable set in R and f be a real-extended function (i.e., f(x) ∈ [−∞, +∞], x ∈ C) defined on C. f is referred to as a Lebesgue measurable function on C or measurable function if for every finite a, the set {x ∈ C : f(x) > a} n is a measurable subset of R , which is often simply denoted as {f > a}. why?

∞ [ C = {f = −∞} ∪ ( {f > −k}) k=1

hC, B(R)i-measurable

In particular, to establish hC, B(R)i-measurability of a map into the real line, it is simplified to check the inverse images of intervals of the form (a, ∞) as follows.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 25 / 35 ∞ [ C = {f = −∞} ∪ ( {f > −k}) k=1

hC, B(R)i-measurable

In particular, to establish hC, B(R)i-measurability of a map into the real line, it is simplified to check the inverse images of intervals of the form (a, ∞) as follows. Definition 4.2 n Let C be a measurable set in R and f be a real-extended function (i.e., f(x) ∈ [−∞, +∞], x ∈ C) defined on C. f is referred to as a Lebesgue measurable function on C or measurable function if for every finite a, the set {x ∈ C : f(x) > a} n is a measurable subset of R , which is often simply denoted as {f > a}. why?

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 25 / 35 hC, B(R)i-measurable

In particular, to establish hC, B(R)i-measurability of a map into the real line, it is simplified to check the inverse images of intervals of the form (a, ∞) as follows. Definition 4.2 n Let C be a measurable set in R and f be a real-extended function (i.e., f(x) ∈ [−∞, +∞], x ∈ C) defined on C. f is referred to as a Lebesgue measurable function on C or measurable function if for every finite a, the set {x ∈ C : f(x) > a} n is a measurable subset of R , which is often simply denoted as {f > a}. why?

∞ [ C = {f = −∞} ∪ ( {f > −k}) k=1

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 25 / 35 Elementary properties of measurable functions

The definition for measurable function is equivalent to any of the following statements hold for finite a: (i) {f ≥ a} is measurable. (ii) {f < a} is measurable. (ii) {f 6 a} is measurable. Corollary 4.1 {f > −∞}, {f < +∞}, {f = +∞}, {a 6 f 6 b}, {f = a}, etc, are all measurable if f is measurable.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 26 / 35 Properties

Theorem 4.1 The finite-valued function f is measurable if and only if f −1(G) is 1 −1 measurable for every open set G of R , and if and only if f (F ) is 1 measurable for every closed set F of R .

Theorem 4.2 1 Let A be a dense subset of R . Then f is measurable if {f > a} is measurable for all a ∈ A.

Remark 4.1 (dense)

A set C ⊂ C1 is said to be dense in C1 if ∀x1 ∈ C1 and  > 0, ∃ a point x ∈ C such that 0 < |x − x1| < . Example: Q is dense in R.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 27 / 35 Theorem 4.3 If f is measurable and if g = f a.e., then g = f a.e., then g is measurable and µ({g > a}) = µ({f > a}).

almost everywhere or a.e.: A property or a statement holds in C except in some subset of C with measure zero. For instance, the statement “f = 0 a.e. in C” is abbreviated of f(x) = 0 in C with the exception of those x in some subset Z of C with µ(Z) = 0.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 28 / 35 almost everywhere or a.e.: A property or a statement holds in C except in some subset of C with measure zero. For instance, the statement “f = 0 a.e. in C” is abbreviated of f(x) = 0 in C with the exception of those x in some subset Z of C with µ(Z) = 0. Theorem 4.3 If f is measurable and if g = f a.e., then g = f a.e., then g is measurable and µ({g > a}) = µ({f > a}).

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 28 / 35 Operations on measurable functions

Theorem 4.4 d If f is continuous on R , then f is measurable. If f is measurable a.e. in C, and Φ is continuous, then Φ ◦ f or Φ(f) is measurable.

Remark 4.2 The cases that arise frequently are

Φ(f) = |f|, |f|p(p > 0), ecf

are measurable if f is measurable. Noting another special case is that of

f + = max{f, 0}, f − = −min{f, 0}

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 29 / 35 Operations on measurable functions

∞ Suppose f and g are measurable, and {fn}n=1 is a sequence of measurable functions, and λ is any real number, then so are {f > g}. f + λ and λf. f + g. fg, and f/g if g 6= 0 a.e.

supn fn(x), infn fn(x), lim supn→∞ fn(x), and lim infn→∞ fn(x). ∞ Plus, if lim fn(x) = f(x) and {fn} = 1 is a collection of measurable n→∞ n functions, then f is measurable.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 30 / 35 Characteristic function or indicator function

Our attention to the objects that lie at the heart of integration theory: measurable functions. The starting point is the notion of a characteristic function or indicator function of a set C, which is defined by ( 1 if x ∈ C χC (x) = 0 if x∈ / C

Clearly, χC is measurable if and only if C is measurable. χC is an n example of what is referred to as a simple function on R : a simple n function on a set C ⊂ R is one that is defined on C and supposes only a finite number of finite values on C. If f is a simple function on C taking distinct values a1, ..., aN on disjoint subsets C1, ..., CN of C, SN and C = k=1 Ck, then

N X f(x) = akχCk (x), x ∈ C. k=1

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 31 / 35 Approximation by simple function

Theorem 4.5 Every function can be represented as the convergence of a sequence {fk} of simple functions. If f ≥ 0, the sequence can be chosen to increasingly converge to f, i.e. fk ≤ fk+1, ∀k.

If f is measurable, then {fk} can be chosen to be measurable.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 32 / 35 Proof

We begin by approximating pointwise, non-negative measurable functions by simple functions. Firstly, start with a truncation within [0, k]. Suppose f ≥ 0, k = 1, 2, ..., subdivide the values of f by partitioning [0, k] into subintervals [(j − 1)2−k, j2−k], j = 1, ..., k2k. Then

( j−1 j−1 j k 2k if 2k ≤ f(x) ≤ 2k , j = 1, ..., k2 fk(x) = k if f(x) ≥ k.

Then, fk(x) → f(x) as k goes to infinity for all x. Clearly, fk ≤ fk+1 −k and fk → f because of 0 ≤ f − fk ≤ 2 as k tends to infinity wherever f is finite and fk = k → +∞ wherever f = +∞. It completes the proof for the second theorem of nonnegative case. In fact,

k2k X j − 1 fk(x) = k χ{ j−1 ≤f(x)≤ j } + kχ{f≥k} 2 2k 2k j=1

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 33 / 35 Approximation by simple function

For the first theorem, using the decomposition of the function f: f = f + − f −. Since both f + and f − are nonnegative, then it trivially yields to apply the above theorem twice. Theorem 4.6 d Suppose f is measurable on R . Then ∃ a sequence of simple function ∞ {fk}k=1 that satisfies

|fk(x)| ≤ |fk+1(x)|, lim fk(x) = f(x), ∀x k→∞

In particular, |fk(x)| ≤ |f(x)|, ∀x, k.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 34 / 35 Theorems of Egorov and Lusin

Theorem 4.7 (Egorov’s Theorem) ∞ Suppose {fk}k=1 is a sequence of measurable functions defined on a measurable set C with µ(C) < ∞ and assume that fk → f a.e. on C. Given  > 0, ∃ a closed set F ⊂ C such that µ(C − F) ≤  and fk → f uniformly on F.

Theorem 4.8 (Lusin) Suppose f is measurable and finite valued on C with µ(C) < ∞. ∀ > 0, ∃ a closed set F with F ⊂ C and µ(C − F) ≤  and such that

f|F is continuous.

Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 35 / 35