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Sorry if this is in the wrong place. I'm trying to understand probability theory a bit more rigorously and so am coming up against things like lebesgue integration and measure theory etc and have a couple of points I haven't quite got my head around.

So starting from the basics, (someone please correct me if I'm wrong on any of this) in contrast to the Riemann integral (I'm very aware I'm not being rigorous here)

[tex] \int_a^b f(x)dx =\lim_{n\to\infty}\sum_i^n f(x_i)(x_{i+1}-x_i) [/tex]

the Lebesgue integral, which exists if the Riemann integral exists, is, for [itex]y=f(x)[/itex],

[tex] \int_{[a,b]} fd\mu =\lim_{n\to\infty}\sum_i^n y_i \mu(x\in [a,b]|y_i\leq f(x)\leq y_{i+1}) [/tex]

where we consider the limit (again not rigorously) of a sum of intervals in therangeof the function for which each value we assign a weight or `measure' which generalises length etc as opposed to the Riemann case where we form intervals on the domain.

So far so good (unless I've made a howler). Now with this second approach I see the point of discussing probability measures (being measures which map to [0,1]) as heuristically we consider, for example, expected values, of for example the function [itex]y=f(x)[/itex], as a sum over the range of a given function such that [itex]\bar{y}=\sum_i y_i P(y_i)[/itex] where [itex]P(y)[/itex] is the probability measure of [itex]y[/itex] or rather, the probability of [itex]f(x)=y[/itex].

Now, taking as an example the Wiener measure [itex]P_W(\omega)[/itex] where now [itex]\omega[/itex] is a set of paths, I keep seeing expectation integrals of the form

[tex]\int_{\Omega}f(\omega)dP_W(\omega)=\bar{f(\omega)}[/tex]

referred to as Lesbesgue type integrals and this I don't understand. This is because [itex]P_W(\omega)[/itex] is the measure of a (set of)path(s). That is [itex]P_W(\omega)[/itex] returns the probability of a set of paths (unless I'm drastically misunderstanding its definition), not the probability of observing [itex]f(\omega)=y[/itex] and they are not the same thing. As such the integral only makes sense to me if the integral is summing over the domain not the range and so is surely not really a Lebesgue integral? As a simpler equivalent consider rolling a number of dice with outcomes [itex]x\in \Omega[/itex] and we want to calculate the expected value of the sum of all of the shown faces, [itex]f(x)=y[/itex]. We can either write this as

[tex]\sum_i y_i P(f(x)=y_i)[/tex]

such that we sum over all possible sums of the shown faces on the dice (eg for two dice [itex]y_i\in \{2,3,4 \ldots 12\}[/itex]) akin to a lesbesgue integral or we can write it as

[tex]\sum_i f(x_i) P(x_i)[/tex]

such that we consider a sum over all events [itex]x_i\in\{\{1,1\},\{1,2\},\ldots,\{6,6\}\}[/itex] which is not akin to a Lebesgue integral yet this is how all expectation values using, for example the Wiener measure, are written.

Further I cannot seem to rationalise this when writing out, in such an example, the explicit path integral as

[tex]\bar{f(\omega)}=\int_\Omega f(\omega)dP_W(\omega)=\int_{\Omega}[\mathcal{D}\omega] f(\omega)e^{-S(\omega)}[/tex]

again [itex][\mathcal{D}\omega][/itex] is constantly referred to as a path integral measure, but as before the integral only makes any sense if you are summing over the domain, ie taking a probability or rather a weight ([itex]e^{-S(\omega)}[/itex]) for everypath, it doesn't have any thing to do with the domain which depends on [itex]f(\omega)[/itex].

I guess this could come down to misunderstanding integration with measure. Am I wrong to think that the summation always has to occur over the range? To clarify I understand that one can always write the integral

[tex] \int_a^b f(x)dx =\int_{[a,b]} fd\mu[/tex],

but this does depend on appropriately defining the measure [itex]\mu[/itex]. What I don't understand is when the measure featuring in the above integral isexplicitlysomething that doesn't depend on [itex]f[/itex], for example, the Wiener measure, which is defined as the probability of observing a set of paths.

Sorry for the rambling! Would be most appreciative if someone could point me in the right direction.

Many thanks,

R

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# Interpreting path integral averages as measure integrals

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