28.4. PROKHOROV AND LEVY THEOREMS 769

denoted by H2j and continuing this way, one can obtain compact subsets of H,

{H i

k

}which

satisfies: each H ij is contained in some H i−1

k , each H ij is compact with diameter less than

i−1, each H ij is the union of sets of the form H i+1

k which are contained in it. Denoting by{H i

j

}mi

j=1those sets corresponding to a superscript of i, it can also be assumed mi < mi+1.

If this is not so, simply add in another point to the i−1 net. Now let{

Iij

}mi

j=1be disjoint

closed intervals in [0,1] each of length no longer than 2−mi which have the property that Iij

is contained in Ii−1k for some k. Letting Ki ≡ ∪mi

j=1Iij, it follows Ki is a sequence of nested

compact sets. Let K = ∩∞i=1Ki. Then each x ∈ K is the intersection of a unique sequence

of these closed intervals,{

Ikjk

}∞

k=1. Define θx ≡ ∩∞

k=1Hkjk. Since the diameters of the H i

j

converge to 0 as i→∞, this function is well defined. It is continuous because if xn→ x, thenultimately xn and x are both in Ik

jk, the kth closed interval in the sequence whose intersection

is x. Hence,d (θxn,θx)≤ diameter(Hk

jk)≤ 1/k.

To see the map is onto, let h ∈ H. Then from the construction, there exists a sequence{Hk

jk

}∞

k=1of the above sets whose intersection equals h. Then θ

(∩∞

i=1Ikjk

)= h. ■

Note θ is probably not one to one.As an important corollary, it follows that the continuous functions defined on any com-

pact metric space is separable.

Corollary 28.4.7 Let H be a compact metric space and let C (H) denote the continuousfunctions defined on H with the usual norm,

|| f ||∞≡max{| f (x)| : x ∈ H}

Then C (H) is separable.

Proof: The proof is by contradiction. Suppose C (H) is not separable. Let Hk de-note a maximal collection of functions of C (H) with the property that if f ,g ∈Hk, then|| f −g||

∞≥ 1/k. The existence of such a maximal collection of functions is a consequence

of a simple use of the Hausdorff maximality theorem. Then ∪∞k=1Hk is dense. Therefore, it

cannot be countable by the assumption that C (H) is not separable. It follows that for somek,Hk is uncountable. Now by Theorem 28.4.6 there exists a continuous function θ definedon a compact subset K of [0,1] which maps K onto H. Now consider the functions definedon K

Gk ≡ { f ◦θ : f ∈Hk} .

Then Gk is an uncountable set of continuous functions defined on K with the propertythat the distance between any two of them is at least as large as 1/k. This contradictsseparability of C (K) which follows from the Weierstrass approximation theorem in whichthe separable countable set of functions is the restrictions of polynomials that involve onlyrational coefficients. ■

The next theorem gives the existence of a measure based on an assumption that a set ofmeasures is tight. It is a sort of sequential compactness result. It is Prokhorov’s theoremabout a tight set of measures. Recall that Λ is tight means that for every ε > 0 there existsK compact such that µ

(KC)< ε for all µ ∈ Λ.