Media type: E-Article Title: A note on Borel’s density theorem Contributor: Furstenberg, Harry imprint: American Mathematical Society (AMS), 1976 Published in: Proceedings of the American Mathematical Society Language: English DOI: 10.1090/s0002-9939-1976-0422497-x ISSN: 0002-9939; 1088-6826 Keywords: Applied Mathematics ; General Mathematics Origination: Footnote: Description: <p>We prove the following theorem of Borel: <italic>If</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>is a semisimple Lie group</italic>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>a closed subgroup such that the quotient space</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper H"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>carries finite measure, then for any finite-dimensional representation of</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>each</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<italic>invariant subspace is</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<italic>invariant</italic>. The proof depends on a consideration of measures on projective spaces.</p> Access State: Open Access