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The $L^{p}$ distance for $0 < p < 1$ is a complete translation-invariant metric

Statement

Let $0 < p < 1$ and let $L^{p} (μ)$ denote the set of almost-everywhere classes of functions in $L^{p} (μ)$ . Define

$d_{p} ([f], [g]) : = \int ∣ f - g ∣^{p} d μ .$

Then $d_{p}$ is a translation-invariant metric on $L^{p} (μ)$ , and $(L^{p} (μ), d_{p})$ is complete.

Facts & Assumptions

Given: A measure space $(X, A, μ)$ and an exponent $0 < p < 1$ .

[L1]

The class notation $L^{p} (μ)$ means almost-everywhere equivalence classes of $L^{p} (μ)$ representatives (The space $L^{p} (μ)$ as the quotient by null functions).

[L2]

A metric and a complete metric space are defined in Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric and Complete metric space: every Cauchy sequence converges in the space.

[L3]

Monotone convergence, dominated convergence, and measurability of pointwise limits are available (Monotone convergence for the integral, Dominated convergence, Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable).

[L4]

Zero nonnegative integral means zero almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[L5]

Sums, scalar multiples, absolute values, and pointwise limits of measurable functions are measurable (Closure properties of measurable functions used by the integral).

[L6]

Countable unions of measurable null sets are measurable and null (Finite and countable subadditivity of measures).

[L7]

For $0 < p < 1$ and nonnegative reals $a, b$ , $(a + b)^{p} \leq a^{p} + b^{p} .$ Indeed, if $a + b = 0$ the claim is trivial. Otherwise set $s : = a + b > 0$ , $u : = a / s$ , and $v : = b / s$ . Then $u, v \in [0, 1]$ and $u + v = 1$ . If $0 < u \leq 1$ , then $\log u \leq 0$ , so $p \log u \geq \log u$ because $0 < p < 1$ ; strict increase of the exponential and the definition of real power therefore give $u^{p} \geq u$ . The same holds for $v$ . Hence $a^{p} + b^{p} = s^{p} (u^{p} + v^{p}) \geq s^{p} (u + v) = s^{p} = (a + b)^{p} .$ (Real powers for positive bases, with the zero-base positive-exponent convention, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The exponential function is strictly increasing) [given]

Proof

Proof technique: Because $(a + b)^{p} \leq a^{p} + b^{p}$ for $0 < p < 1$ , $d_{p} ([f], [g]) = \int ∣ f - g ∣^{p} d μ$ defines a metric on quotient classes and is translation invariant. Completeness follows by repeating the Riesz-Fischer telescoping argument without taking $p$ -th roots.

1.1

If $f \sim f^{'}$ and $g \sim g^{'}$ , then $f - f^{'}$ and $g - g^{'}$ vanish almost everywhere. Outside the union of those two null sets, one has $f - g = f^{'} - g^{'}$ , so $∣ f - g ∣^{p} = ∣ f^{'} - g^{'} ∣^{p}$ almost everywhere. Hence $d_{p} ([f], [g])$ is well defined. The same union-of-null-sets argument shows that addition and scalar multiplication descend to the quotient classes, and the inequality in [L7] shows that $L^{p} (μ)$ is closed under those operations.

L1L5L6L7

1.2

Symmetry of $d_{p}$ is immediate. If $d_{p} ([f], [g]) = 0$ , then $\int ∣ f - g ∣^{p} d μ = 0$ , so $f = g$ almost everywhere and hence $[f] = [g]$ . For the triangle inequality, the pointwise inequality from [L7] gives [L2, L4, L7] $∣ f - h ∣^{p} = ∣ (f - g) + (g - h) ∣^{p} \leq ∣ f - g ∣^{p} + ∣ g - h ∣^{p},$ and integrating yields $d_{p} ([f], [h]) \leq d_{p} ([f], [g]) + d_{p} ([g], [h]) .$ Thus [L2] makes $d_{p}$ a metric.

1.3

Let $(u_{n})$ be Cauchy in $d_{p}$ . Choose by least indices a subsequence $(u_{n_{k}})$ with [L3, L5, given, choose] $d_{p} (u_{n_{k + 1}}, u_{n_{k}}) < 2^{- k} .$ Choose representatives $f_{k}$ of $u_{n_{k}}$ and define $h_{k} : = ∣ f_{k + 1} - f_{k} ∣^{p}, g_{m} : = \sum_{j < m} h_{j} .$ Each $h_{k}$ is measurable and integrable, and [L3] gives a measurable pointwise limit $g : = \sum_{j = 0}^{\infty} h_{j}$ with $\int g d μ = \lim_{m \to \infty} \int g_{m} d μ \leq \sum_{j = 0}^{\infty} 2^{- j} < \infty .$ Hence $g < \infty$ almost everywhere.

2.1

Translation invariance is pointwise: [step 1.1] $d_{p} ([f] + [u], [g] + [u]) = \int ∣ (f + u) - (g + u) ∣^{p} d μ = d_{p} ([f], [g]) .$

2.2

Fix $x$ outside the null set where $g (x) = \infty$ . Then $\sum_{j} ∣ f_{j + 1} (x) - f_{j} (x) ∣^{p} < \infty$ , so the terms tend to $0$ . Thus $∣ f_{j + 1} (x) - f_{j} (x) ∣ \leq 1$ for all large $j$ , and then [step 1.3, L3, L7] $∣ f_{j + 1} (x) - f_{j} (x) ∣ \leq ∣ f_{j + 1} (x) - f_{j} (x) ∣^{p} = h_{j} (x) .$ So the real series $\sum_{j} ∣ f_{j + 1} (x) - f_{j} (x) ∣$ converges by comparison with $\sum_{j} h_{j} (x)$ , which makes $(f_{k} (x))$ converge to some real value $f (x)$ . By [L3], the resulting function $f$ is measurable. Also $∣ f (x) - f_{k} (x) ∣^{p} \leq {(\sum_{j \geq k} ∣ f_{j + 1} (x) - f_{j} (x) ∣)}^{p} \leq \sum_{j \geq k} h_{j} (x) .$ Integrating and using monotone convergence on the tails yields $d_{p} (u_{n_{k}}, [f]) \leq \sum_{j \geq k} d_{p} (u_{n_{j + 1}}, u_{n_{j}}) \leq \sum_{j \geq k} 2^{- j} \to 0.$

3.1

Because $(u_{n})$ is Cauchy, given $ε > 0$ choose $K$ with $d_{p} (u_{n}, u_{m}) < ε / 2$ for $m, n \geq K$ , then choose $k$ with $n_{k} \geq K$ and $d_{p} (u_{n_{k}}, [f]) < ε / 2$ from step 2.2. The triangle inequality from step 1.2 gives $d_{p} (u_{n}, [f]) < ε$ for all $n \geq K$ . Hence $(L^{p} (μ), d_{p})$ is complete.

step 1.2step 2.2

4.1

Steps 1.2 and 2.1 prove that $d_{p}$ is a translation-invariant metric, and step 3.1 proves completeness.

step 1.2step 2.1step 3.1∎

Depends on

Used by

The p-power triangle inequality for nonnegative functions when 0 < p < 1 Theorem

Dependency tree · two levels

59 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Theorem 8.16 (standard reference, not scraped)
John K. Hunter, Measure Theory, reverse inequality discussion before Definition 7.6 (standard reference, not scraped)

The LpLp distance for 0<p<10<p<1 is a complete translation-invariant metric