Example

Tag 08LO, permanent. Cite it as [Bourbaki, Tag 08LO].

Integration, chapter III, § 1. Measures on a locally compact space

— 1) The measure $\varepsilon_a$ defined by a unit mass at a point $a \in X$ is bounded, and $\|\varepsilon_a\| = 1$ .

2) The Lebesgue measure on $\mathbf{R}$ is not bounded; indeed, for every integer $n > 0$ there exists a function $f \in \mathscr{K}(\mathbf{R}; \mathbf{C})$ with values in $[0, 1]$ and equal to 1 on the interval $[ -n, n ]$ (No. 2, Lemma 1); thus $\|f\| = 1$ and

\int_{-\infty}^{+\infty} f(x)\, dx \geq \int_{-n}^{n} f(x)\, dx = 2n ,

which proves that there does not exist any finite number M satisfying the relation (19).

3) On the real line $\mathbf{R}$ , the mapping

f \mapsto \int_{-\infty}^{+\infty} \frac{f(x)\,dx}{1 + x^2}

is a bounded measure since, for every function $f \in \mathscr{H}(\mathbf{R}; \mathbf{C})$ ,

\left| \int_{-\infty}^{+\infty} \frac{f(x)\,dx}{1 + x^2} \right| \leq \|f\| \int_{-\infty}^{+\infty} \frac{dx}{1 + x^2} = \pi \cdot \|f\|.

Since the relations $\|f\| \leq 1$ and $\|\overline{f}\| \leq 1$ are equivalent, it follows from (20) that

(21)

\|\overline{\mu}\| = \|\mu\|

for every measure

\mu

X

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

No. 2, Lemma 1