Dear All, It sounds a strange question, we know that the measure theory is the modern theory while the topological spaces is the classical analysis (roughly speaking). And measure theory solves some problems in the classical analysis. My first question is that right? Second, Is every measurable space a topological space? Any more explanation will be appreciated. Thanks in advance
No, not at all. You make it seem that a measure space is some kind of updated version of a topology. This is totally false. Measure spaces and topological spaces are not comparable to each other as each can be useful in its own right: Topological spaces are used to define a notion of "closeness". With it, you can intuitively speak about points which are close to each other. (However, we may not know how close: this is a metric space). A topological space is essential in geometry. It is used everywhere in algebraic geometry and differential geometry. Furthermore, it is also very much used in analysis. You can't study functional analysis (for example) without knowing your topology. A measure space serves an entirely different goal. A measure space is made to define integrals. Indeed, it is the least information you need in order to be able to integrate. If you want to define length, area, etc. then you need a measure space. As you can expect, a measure space is useful everywhere where integrals are used. Note that most applications of measure theory already have a topology. The measure space will then be defined on the topological space: the borel sets will be generated by the open sets. For example, when studying groups, we often look at locally compact topological groups. It is possible to define a canonical measure space on such a group. This is useful in harmonic analysis. No. Measure spaces and topological spaces have nothing to do with each other. It is possible to have a measurable space without a topology (this will not be very useful though). However, I think it's fair to say that most applications of measurable spaces do carry a topology.
One variant of measure theory is probability theory (total measure = 1). Here topology seldom plays a role, since the probability space is usually defined completely abstractly.
Thank you for these explanations. May be we know some theory of measure and some topological spaces, but the problem is that we (at least I ) do not know why do we use this theory. I do not know if there are some aspects from this angle, not only theorems with their proof! For example, what is topology, what is the relation with geometry.
To understand that, you must study geometry. Topology is a natural framework for studying things like continuity, compactness, connectedness,... I'd suggest picking up a differential/algebraic topology book to see the connections.
Don't we need topology to show countable additivity? However to show that the measure is countably additive do we not need some kind of topological argument? Akin to heine-borel theorem? Ramesh
Topology is mostly about global aspects, and geometry is more into local aspects of your space. You can use generalizations like uniform spaces to have a notion of closeness. A more general definition of closeness could be something like topological distinguishability of points, tho this does not give you too- clear -of -a -picture in this respect. Why not try to see if the collection of sets in your space satisfy the axioms of a topological space?
One distinction between geometry and topology is that the spaces studies under geometry have a distance relationship between points, while topology only uses the concepts of open and closed sets.