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A locally finite nonnegative family with positive pointwise sum normalizes to a partition of unity

Statement

Let ${f_{s} : X \to [0, \infty)}_{s \in S}$ be continuous with locally finite cozero family, and suppose $f : = \sum_{s} f_{s}$ is positive at every point. Then $φ_{s} : = f_{s} / f$ form a partition of unity; their cozero sets and supports are the same as those of the corresponding $f_{s}$ .

Facts & Assumptions

Given: A locally finite nonnegative continuous family whose pointwise sum is everywhere positive.

[L1]

The sum $f = \sum_{s} f_{s}$ is continuous (A locally finite family of continuous nonnegative functions has a continuous pointwise sum).

[L2]

A quotient of continuous real-valued maps is continuous on the cozero set of its denominator (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined).

[F1]

A family of continuous maps $X \to [0, 1]$ is a partition of unity exactly when its cozero family is locally finite and its pointwise sum is one (Locally finite partitions of unity and subordination to an open cover).

Proof

technique · direct

1.1L1

By [L1] the function $f$ is continuous, and the positivity hypothesis makes $coz (f) = X$ .

2.1step 1.1L2

Therefore each $φ_{s} = f_{s} / f$ is continuous by [L2] and nonnegative. Since $f_{s} (x) \leq f (x)$ , it takes values in $[0, 1]$ , and positivity of $f$ gives $coz (φ_{s}) = coz (f_{s})$ .

3.1step 2.1

At every $x \in X$ , local finiteness makes the sum finite and gives $\sum_{s} φ_{s} (x) = \sum_{s} f_{s} (x) / f (x) = f (x) / f (x) = 1$ .

4.1step 2.1step 3.1F1∎

The cozero family is unchanged, hence locally finite, and equality of cozero sets also gives equality of supports. Thus [F1] says that ${φ_{s}}$ is a partition of unity.

A locally finite nonnegative family with positive pointwise sum normalizes to a partition of unity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources