Light scattering by particles is often used to determine velocities or concentrations of particles in gaseous or liquid streams. Within the Lorenz-Mie theory, light scattering is well understood both for a single compact spherical particle and a single multilayered particle in a non-absorbing surrounding medium. However, in some cases of practical importance the Lorenz-Mie theory in its present form may fail to describe the scattering because the host medium is absorbing (e.g. water droplets in oil). In this case, a new treatment of the scattering theory is required. In previous work, solutions were obtained in the far-field of the scattering sphere. In this paper, a rigorous solution is derived from the calculation of the total absorption rate of the particle in the host medium, which is valid for all distances from the surface of the encapsulated particle. It is shown that it is necessary to consider finite sizes R of the integrating sphere when dealing with absorbing host media. Cross-sections are defined which are characteristic quantities not only for the particle, depending on the size of a conceptual sphere around the scatterer and the imaginary part of the refractive index of the host medium. The results obtained are discussed for the case of non-absorbing host media and in the far-field approximation. Some numerical examples are given which are also related to experimental results.