On convergent subsequences of functions that are locally of bounded total variation
In mathematics, Helly's selection theorem (also called the "Helly selection principle") states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence.
In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions.
It is named for the Austrian mathematician Eduard Helly.
A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point.
The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures.
Statement of the theorem.
Let ("f""n")"n" ∈ N be a sequence of increasing functions mapping a real interval I into the real line R,
and suppose that it is uniformly bounded: there are "a,b" ∈ R such that "a" ≤ "f""n" ≤ "b" for every "n"  ∈  N.
Then the sequence ("f""n")"n" ∈ N admits a pointwise convergent subsequence.
Proof.
Step 1. An increasing function "f" on an interval I has at most countably many points of discontinuity..
Let formula_1, i.e. the set of discontinuities, then since "f" is increasing, any "x" in A "satisfies" formula_2, where formula_3,formula_4, hence by discontinuity, formula_5. Since the set of rational numbers is dense in R, formula_6 is non-empty. Thus the axiom of choice indicates that there is a mapping "s" from A to Q. 
It is sufficient to show that "s" is injective, which implies that A has a non-larger cardinity than Q, which is countable. Suppose "x1,x2"∈A, "x1" ;"x2", then formula_7, by the construction of "s", we have "s(x1)2)". Thus "s" is injective. 
Step 2. Inductive Construction of a subsequence converging at discontinuities and rationals..
Let formula_8, i.e. the discontinuities of "f""n," formula_9, then A is countable, and it can be denoted as {"an": "n"∈N}.
By the uniform boundedness of ("f""n")"n" ∈ N and B-W theorem, there is a subsequence ("f(1)""n")"n" ∈ N such that ("f(1)""n""(a1)")"n" ∈ N converges. Suppose ("f(k)""n")"n" ∈ N has been chosen such that ("f(k)""n""(ai)")"n" ∈ N converges for "i"=1...,k, then by uniform boundedness, there is a subsequence ("f(k+1)""n")"n" ∈ N of ("f(k)""n")"n" ∈ N, such that ("f(k+1)""n""(ak+1)")"n" ∈ N converges, thus ("f(k+1)""n")"n" ∈ N converges for "i"=1...,k+1.
Let formula_10, then g"k" is a subsequence of "f""n" that converges pointwise in A.
Step 3. "gk" converges in I except possibly in an at most countable set..
Let formula_11, then , "hk(a)=gk(a)" for "a"∈A, "hk" is increasing, let formula_12, then h is increasing, since supremes and limits of increasing functions are increasing, and formula_13 for "a"∈ A by Step 2. By Step 1, "h" has at most countably many discontinuities.
We will show that "gk" converges at all continuities of "h". Let "x" be a continuity of "h", "q,r"∈ A, "q ;x ;r", then formula_14,hence
formula_15
formula_16
Thus,
formula_17
Since "h" is continuous at "x", by taking the limits formula_18, we have formula_19, thus formula_20
Step 4. Choosing a subsequence of "gk" that converges pointwise in I.
This can be done with a diagonal process similar to Step 2.
With the above steps we have constructed a subsequence of ("f""n")"n" ∈ N that converges pointwise in I.
Generalisation to BVloc.
Let "U" be an open subset of the real line and let "f""n" : "U" → R, "n" ∈ N, be a sequence of functions. Suppose that
("f""n") has uniformly bounded total variation on any "W" that is compactly embedded in "U". That is, for all sets "W" ⊆ "U" with compact closure "W̄" ⊆ "U",
formula_21
where the derivative is taken in the sense of tempered distributions.
Then, there exists a subsequence "f""n""k", "k" ∈ N, of "f""n" and a function "f" : "U" → R, locally of bounded variation, such that
formula_22 ;ref name="10.1093/oso/9780198502456.001.0001" ; ;/ref ;
formula_23
Further generalizations.
There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaces, is due to Barbu and Precupanu:
Let "X" be a reflexive, separable Hilbert space and let "E" be a closed, convex subset of "X". Let Δ : "X" → [0, +∞) be positive-definite and homogeneous of degree one. Suppose that "z""n" is a uniformly bounded sequence in BV([0, "T"]; "X") with "z""n"("t") ∈ "E" for all "n" ∈ N and "t" ∈ [0, "T"]. Then there exists a subsequence "z""n""k" and functions "δ", "z" ∈ BV([0, "T"]; "X") such that
formula_24
formula_25
formula_26
References.
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