Brief Look into Measure Theory 🌱
A sigma algebra is a special collection of specific subsets of a set that satisfy 3 following properties:
We can then define a measure map as the following:
Here are some examples of measures:
We can say a function mapping is measurable if the following condition holds:
Lebesgue Integral
Problems with the Reimann Integral
Reimann vs Lebesgue Integral
Defining the Lebesgue Integral
For simple (step, staircase, …) functions the Lebesgue Integral is defined as follows (where is some function mapping to the real numbers and :
Generally, it is defined as follows:
The integral of a measurable function is the supremum of the set of all integral values for step functions that lie below the function .
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