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Welcome to lecture twenty-five on measure and integration. In the previous lectures we had started looking at measure and integration on product spaces. In the previous, we defined the notion of product sigma algebra and today, we will define the notion of product measure. . So, let us recall. We will fix for todays discussion 2 measure spaces, X A mu and Y B nu. So, X is a set, A is a sigma algebra of subsets of X and mu is a measure defined on the sigma algebra A. And similarly, for the measure space Y B nu, B is sigma algebra of subsets of Y and nu is a measure on the sigma algebra B. .. So, we have already defined the notion of the product measure, namely A cross B. So, if you recall, so we defined the notion of A times B, so this is the sigma algebra generated by all rectangles and rectangles were defined as the sets A times B, where A belongs to the sigma algebra A and B belongs to the sigma algebra B. So, now we have given a measure mu on the sigma algebra A and given a measur