Mean values using eigenvalues - how?

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I thought the definition of the average value of $f(x)$ was $f* = \frac{1}{T}\int_0^T f(t)dt$

How do I get from this definition to the definition of the mean value of a function in terms of eigenvalues?

The mean value of a function is the sum of all the eigenvalues, each multiplied by a probability...