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Measures Integrals and Martingales 2nd Edition PDF Form

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[Music] i think most of us are familiar with the usual way of calculating integrals let's consider a function f of x if i want to calculate the integral over some bound which we typically think as the area under the curve we divide the region into vertical rectangles and sum up the areas of each of these rectangles to get the final integral we consider smaller and smaller rectangles and the integral is the sum of the areas of all of these rectangles as the number of rectangles approaches infinity the sum of rectangles is called riemann sums so this technique is called riemann integration however this technique definitely has its problems the first shortcoming of the riemann integration is the difficulty to extend this intuition to higher dimensions to approximate the volume under a surface we would use cuboids rather than rectangles but if we move to four dimensions we would use well we don't know visualizing a four-dimensional object is incredibly difficult the second and most importa

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