Measuring a set

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                  Measuring is one of the basic and fundamental thing in mathematics.

                  From our childhood we learn how to measure any natural things, like if we count something that also kind of measuring, if we buy some thing that purchase quantity is also kind of a measure, similarly we learn in geometry many things like length of a line, area of a plane, volume of a object and using this result and using the concept of integration we can find the length of curved line, area of a surface, volume of a object. All the above we discuss all are a kind of measurement.

                 This above result gives us a very important idea about the object we considering and gives about the knowledge of different results. So when we see generally it is very important for us to have a general knowledge of measurement for a general set. That give us more knowledge about that set.

                So mathematicians introduce the notion of measure theory, that give us a general idea how to measure any set.

                But defining everything we have to keep in mind that existing results are should not nullified and whatever the new method we introduce that our existing results sit perfectly there.

                 Now if I take any arbitrary set and try to measure that set that will be no meaning or we can say we don't have any idea about set so the problem is where I will began and what I will take to work, if I take something particular as a result my set become a particular set, its not an arbitrary set then.

               So its important for us to define some kind of set with some special property and continue our study on that. Obviously we should make the property such a way that its rigorous and we have lots of real example on that.

               So the idea is what kind of property that set I should give, that satisfy my natural intuition also.


        Here is some kind,


               Algebra :- Let X be an arbitrary set. A collection  A of subsets of X is called an algebra of subsets of X if it satisfies the following conditions:

     1) X ∈ A.

     2) A ∈ A implies that Ac ∈  A.

     3) A and B ∈ A implies A ∪ B ∈  A.


           Sigma-Algebra :- Let X be an arbitrary set. A collection S of subsets of X is called an sigma-algebra of subsets of X if it satisfies the following conditions:


     1) X ∈ S.


     2) A ∈ S implies that Ac ∈ S.


     3) An ∈ S for n ∈ N implies ∪An ∈ S.


            

                     Now we are in a position to define measure formally,

                    Measure:- If S be a sigma-algebra of subsets of a set X. A set function u defined on S is called a measure if it satisfies the following conditions,

   1) nonnegative extended real-valued i.e. u(E) ∈ [0,∞] for all E ∈ S.

   2) u(Φ)= 0

   3) Countable Additivity: u(∪ En) =  Σ u(En)


So up to now we get some proper defining collection that we name as algebra, sigma-algebra and we define measure on sigma-algebra.








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