- #1

cappadonza

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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 [tex] F : \mathcal{F} \rightarrow \mathbb{R} [/tex] where [tex] \mathcal{F} [/tex] is a family of sets.

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

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 in to this.

sorry if this is a bit vague, i'm trying to get my head around this stuff

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

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 in to this.

sorry if this is a bit vague, i'm trying to get my head around this stuff

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