The Lebesgue Integral
The Lebesgue integral extends the Riemann integral to wider classes of functions. The definition of Lebesgue integral is a little bit more complicated, but the properties are much more convenient in applications. That is why the modern mathematics mainly uses the Lebesgue integral. To strengthen the analogy with the Riemann integral, in this book the theory of the Lebesgue integral is treated on the basis of Fréchet’s definition, namely, by means of the convergent integral sums analogous to the Riemann integral sums. One of the advantages of this approach is the simplicity with which this definition extends to vector-valued functions.