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Uniform-on-compacts metric on continuous path space

Definition

Let $C = C ([0, \infty), R)$ be the set of continuous real-valued paths. For $f, g \in C$ define $d_{u o c} (f, g) = \sum_{n = 1}^{\infty} 2^{- n} (1 \land \max_{0 \leq t \leq n} ∣ f (t) - g (t) ∣) .$ Then $d_{u o c}$ is a finite metric on $C$ , and its metric topology is exactly the topology of uniform convergence on compact subsets of $[0, \infty)$ . We call it the uniform-on-compacts metric.

Facts & Assumptions

Given: Continuous paths $f, g, h \in C$ .

[F1]

The compact-convergence topology has basic neighborhoods requiring uniform closeness on one compact set. The topology of compact convergence on $C (X, Y)$ for metric $X$ and $Y$ : uniform convergence on each compact subset of $X$

[F2]

A metric is symmetric, separates points, and satisfies the triangle inequality. Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric

[F3]

Closed bounded real intervals are compact, and a continuous real function on a nonempty compact metric space attains a finite maximum. Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value

[F4]

The geometric series satisfies $\sum_{n = 1}^{\infty} 2^{- n} = 1$ , and its tail $2^{- N}$ tends to zero. For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$

[F5]

Every real bound is exceeded by an integer. Every complete ordered field is Archimedean

[F6]

The paths under discussion are continuous maps. Continuity of a map of topological spaces at a point and globally

Verification

technique · direct

1.1

By [F3] and [F6], each maximum in the definition exists and is finite. Every summand lies in $[0, 2^{- n}]$ , so [F4] proves that the series converges to a finite value in $[0, 1]$ .

F3F4F6

1.2

Let $K \subseteq [0, \infty)$ be compact and let $ε > 0$ . If $K$ is empty, its basic neighborhood from [F1] is all of $C$ . Otherwise the identity function attains a finite maximum on $K$ by [F3], and [F5] gives an integer $N \geq 1$ with $K \subseteq [0, N]$ . Put $a = \min (1 / 2, ε)$ . If $d_{u o c} (f, g) < 2^{- N} a$ , then the $N$ th term gives $1 \land \max_{t \leq N} ∣ f (t) - g (t) ∣ < a,$ hence $∣ f - g ∣ < ε$ throughout $K$ . Thus every compact-convergence basic neighborhood contains a $d_{u o c}$ -ball.

F1F3F5algebra

1.3

Conversely, given a $d_{u o c}$ -ball of radius $ε > 0$ , use [F4] to choose $N$ with $2^{- N} < ε / 2$ . If $\max_{t \leq N} ∣ f (t) - g (t) ∣ < ε / 2$ , then the first $N$ terms sum to less than $ε / 2$ , while [F4] makes the remaining tail at most $2^{- N} < ε / 2$ . Hence $d_{u o c} (f, g) < ε$ . The neighborhood controlling the compact interval $[0, N]$ therefore lies inside the metric ball.

F1F3F4algebra

2.1

Symmetry is termwise. If $d_{u o c} (f, g) = 0$ , every nonnegative summand is zero; hence $f = g$ on every $[0, n]$ , and therefore on their union $[0, \infty)$ . Conversely $f = g$ makes every term zero.

step 1.1F2algebra

3.1

For each $n$ , the ordinary triangle inequality gives $\max_{t \leq n} ∣ f (t) - h (t) ∣ \leq \max_{t \leq n} ∣ f (t) - g (t) ∣ + \max_{t \leq n} ∣ g (t) - h (t) ∣ .$ Since $1 \land (a + b) \leq (1 \land a) + (1 \land b)$ for $a, b \geq 0$ , multiplication by $2^{- n}$ and summation give $d_{u o c} (f, h) \leq d_{u o c} (f, g) + d_{u o c} (g, h)$ . With step 2.1, [F2] proves that $d_{u o c}$ is a metric.

step 1.1step 2.1F2algebra

4.1

Steps 1.2 and 1.3 give mutual refinement of the neighborhood bases at every $f$ , so the two topologies coincide. Equivalently, $d_{u o c} (f_{j}, f) \to 0$ exactly when $f_{j} \to f$ uniformly on every compact subset. No choice is used: maxima and integer bounds exist with unique least choices if a witness is desired, and every sum is over the fixed natural order.

step 1.2step 1.3F1∎

Source notes

The bounded weighted-sum metric is the standard metrization of local uniform convergence. The verification records both neighborhood containments, including the empty compact set and the geometric tail.

Depends on

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Sources

van der Vaart and Wellner, Weak Convergence and Empirical Processes, Sections 1.3 and 1.5 (standard reference, not scraped)