:Have you looked at our c�dl�g and Borel functions articles?
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The spectral measures can be used to extend the continuous functional calculus to bounded Borel functions.
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The starting point of the proof is the imbedding of X into the Banach space of Borel functions on X, equipped with the sup norm.
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Are there any c�dl�g functions which aren't also Borel functions ?-- talk ) 18 : 03, 30 October 2011 ( UTC)
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Degrees given by Lipschitz functions are called " Lipschitz degrees ", and degrees from Borel functions " Borel-Wadge degrees ".
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More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function.
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As to the general definition of a matrix f ( A ) or f ( A, B ) & c, as function of one or more matrices, note that we do it also with operators in place of matrices, or even more generally, in abstract Banach algebras; the function f need not to be analytic, nor even continuous, but only a borel function may be sufficient, depending on the context.
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