Vol. 23, No. 2, 1967

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Measures whose range is a ball

Neil William Rickert

Vol. 23 (1967), No. 2, 361–371

It has been shown by R. P. Kaufman and the author that if μ is a measure of total variation 1 with values in Rn, then there is a measurable set E with

        -1-----Γ (n∕2)-
|μ(E)| ≧ 2π1∕2Γ ((n + 1)∕2).

The main purpose of this paper is to determine for which measures μ there is no set E with

|μ(E)| > -1-----Γ (n∕2)-.
2π1∕2Γ ((n + 1)∕2)

It will be shown that they are the measures which satisfy the following two conditions:

(i) The measure of the whole space is zero.

(ii) The induced probability measure α f(|μ|) on the projective space Pn1 is orthogonally invariant, where f = dμ∕d|μ| maps the measure space to the sphere Sn1 and α is the natural map of Sn1 onto Pn1.

Mathematical Subject Classification
Primary: 28.50
Received: 22 August 1966
Revised: 3 May 1967
Published: 1 November 1967
Neil William Rickert