MATH MAANIAC

         In my last blog on "Measuring a set", we see that how we can generalized the idea of length, area, volume for an set and we define special kind of set notions like sigma-algebra, algebra and define an arbitrary measure on that set we called that "Measure of a set."      As beginning of the last blog I told that our generalization should be same as our existing idea of length, area, volume. The  Lebesgue Measure on  ℝ  is a generalization of length.            So there is two question, 1) Any subset of   ℝ  we take, can we measure it ? 2) The measure of the set and the length of the set is same ?        We know that, if   ℝ  be my set then P( ℝ ) is the set of all subsets of  ℝ . Let's see we can measure any element of P( ℝ ) or not. To do that we first define equivalence definition of Lebesgue Measure on  ℝ .     Lebesgue Measure on    ℝ     Let E be any subset of  ℝ , we consider the set T = {  Φ  or set of open intervals in  ℝ }, if we can cover my se

                  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