What is the relationship between integrals and measures in measure theory?

[cappadonza]

I just beginning to study measure theory. so far from what i understand so far , can we say in general, an integral is a measure, (ie it is nothing but a set function. a mapping $$F : \mathcal{F} \rightarrow \mathbb{R}$$ where $$\mathcal{F}$$ is a family of sets.

does it make sense to say in general an integral of a function F, is $$\int_{A} F d\mu$$ is the measure of the image of the $$F$$ over some set $$A$$ using the measure $$\mu$$. with the condition the image of $$F$$ over $$A$$ must be measurable using the measure $$\mu$$

so for example the two that i know are lebesgue-integral, lebesgue-stieltjes integral, are basically are general integrals using different measures.

if we could say the above then where would the riemann integral fit into this.
sorry if this is a bit vague, I'm trying to get my head around this stuff

 

[g_edgar]

No, I would say your description is wrong. Why not wait a couple more weeks in the course, then try to fomulate it again?

 

[cappadonza]

I'm not actually doing a course, i work full-time, its something I'm trying to learn through self-study. i go back re-learn what i thought i knew and re-formulate it in the next week or so
thanks

 

[cappadonza]

okay here is my second attempt:
A measure is a set function $$\mathcal{F} \to \Re$$. where $$\mathcal{F}$$ is a sigma-algebra. the invariants it must satisfy are it countable additive and the measure of a null-set is zero.
Now integral $$\int_{B} f d\mu$$ is nothing but a special case of a measure, where it calculates the measure of the set described by $$f$$ over the set $$B$$. The arbitrary measure $$\mu$$ is use to calculate the measure of this set.
Since we viewing integral is as measure it must satisfy the invariants of a measure such as being countable additive. The Riemann-integral only satisfies this condition only a small class of functions, this why the Lebesgue integral is introduced, to over come some of these shor-coming

 

[winterfors]

I'd say you're on the right track.

Even though the integral of a real-valued (measurable) function over a subset D of the domain on which the function is defined is not in general a measure, it is certainy possible (and easy) to define functions whose integral will be a measure.

E.g. a strictly positive real-valued function will have an integral that is a measure, but the intgral of a sine-wave will not fulfill the axioms of a measure.

You can, in fact, view any real-valued positive function as a quotient between two different measures on the same sigma-algebra: see the "Radon–Nikodym theorem", it was of great help for me understanding measures in relation to measurable functions.