Tensor Integral: A New Comprehensive Approach to the Integration Theory

We use tensor product to introduce a new approach to the theory of integration. Such an approach will strengthen the existing various classical concepts of integral and will provide a continuous thread tying the subject matter together. The integral of vector-valued functions with respect to vector-valued additive measures will be covered without any assumption of measurability. As applications, we state and prove extensions of the Lebesgue fundamental theorems of convergence in a more general setting.