Radó's colloquium is a systematic treatment of Lebesgue theory, with an emphasis on the work of Morrey and of Radó and his students, especially in two dimensions. At the time, there were important current problems surrounding Lebesgue's theory for parameterized and unparameterized surfaces, which the book addresses. The exposition begins with reviews of Lebesgue integration and relevant topics in topology, including Fréchet equivalence, the approximation of monotone maps by homeomorphisms, Peano spaces, and a discussion of the topological index of maps into the plane. After a development of further ideas and tools from topology and measure theory, Radó addresses an essential question that equates two sorts of areas for surfaces represented by maps of a 2-cell or a 2-sphere into 3-space.