Can the Measure Be Obtained from the Value of an Integral?

[Kevin_spencer2]

Let be a Lebesgue integral with a measure M on the interval (a,b) so:

$$\int_{a}^{b}dMf(x)=I(a,b,M)$$

We don't know or can't say what M (measure) is , however my question is if we had a trial function U(x) so we could calculate I(a,b,M) for this U without recalling to the measure,either by numerical or other methods my question is if we could obtain the form of the measure , from the value of the integral:

$$\int_{a}^{b}dM\mathcal U(x)$$ ?.

I mean, to know the lebesgue integral you should know the measure, but if you knew the exact (or approximate) value of an integral could you extract the measure from it?.

 

[HallsofIvy]

A single function over a fixed interval? No. The measure has to be defined over each measurable set. Of course, it you knew the integral of f(x)= 1 over every measurable set, that would be the measure.

 

[Kevin_spencer2]

could we use the 'aximatic' result due to dirac delta function, or dirac measure? so:

$$\int_{a}^{b}dM\delta (x-c)=1$$ with a<c<b (or a similar result)

I would be interested mainly in infinite-dimensional case so:

$$\int \mathcal [\phi] \delta (\phi) =1$$ or similar.

 

[StatusX]

If a,b are arbitrary, then this allows you to define a measure by setting:

$$\mu((a,b))=\int_{a}^{b}dMf(x)$$

Then, since any open set is the countable disjoint union of open intervals, this extends uniquely to a measure on the open sets, and so also on the borel sets (the sigma algebra generated by the open sets).

This can be done for an arbitrary complex function f, although if f is not a non-negative function, what you get is not an ordinary positive measure, but what's called a complex measure. If f=1, then this recovers the lebesque measure (or whatever dM is).

 

[Kevin_spencer2]

thankx and a last question, if you have the multi-dimensional integral:

$$\int_{V}d\mathcal M f(X)$$

where M is the measure and it's known then my question is how could you solve this integral by Numerical methods?, what happens in the infinite-dimensional case with a 'Gaussian measure' (the only that can be defined in a inifinite dimensional space) now that you have the measure and the integrand could you evaluate it by numerical metods?? even in the case that the space is infinite-dimensional (i.e function space)