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In every analysis course, one learns some integrals of heavily used function -- e. g. polynomials, exp, sin etc. -- with respect to (w. r. t.) the Lebesgue measure.

Is there anything similar for the Lebesgue integral w. r. t. an arbitrary (finite) measure? I do know the Integral Operator's linearity and the dominated convergence theorem (and Convolution of two probability distributions and other stuff from a measure theory lecture).

In other words, is there something like this, perhaps in form of a table, for often used functions like polynomials, exp, sin etc. (restricted to an interval $[a, b]$ with appropriate scaling factor) and often used measures like Angle Measure, Gaussian Measure?

This may be a broad question, but nevertheless, it's interesting. I'm sure we learnt a theorem to calculate integrals w. r. t. probability measures by transforming it to an integral w. r. t. the Lebesgue measure (in a lecture foregoing measure theory), but I do not know the concise english word for this operation. It may be that it's only a matter of scaling, the described operation and perhaps a generalization to $\sigma$-finite measures by decomposing into countable many subsets on which the measure is finite (Is there a technical term for this procedure?).

Many Thanks in advance.

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  • $\begingroup$ I haven't really seen such a thing. Frequently exact values are irrelevant - we only need estimates. For measures which are just multiples of Lebesgue measure we can frequently change variables, and for others... well, the structure can be arbitrarily complicated and exact values are both hopeless and meaningless. $\endgroup$ – user296602 Oct 7 '18 at 22:17
  • $\begingroup$ If your measure $\mu$ is absolutely continuous w.r.t. Lebesgue measure with density $h$, then $$\int f(x) \; d\mu(x) = \int_{-\infty}^\infty f(x) h(x) \; dx$$ So this depends just as much on $h$ as it does on $f$, and an ordinary table of integrals will suffice (at least if $h$ is a "nice" function). $\endgroup$ – Robert Israel Oct 7 '18 at 22:41

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