Abstract
Still stronger and more important than (b) is strict ergodicity. We suppose m(Q) to be finite. (c) Let f(P) be an arbitrary m-summable function on Q. The time-average off(P) along a curve of motion is then,in general, equal to fj(P)dm/m(Q), the exceptional curves forming a point set on Q of m-measure zero. How these concepts are interrelated is seen most clearly if we state them in the following way. (a') Every open point set on Q that is invariant under the flow is everywhere dense on Q.
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