Probability theory based on measure theory does not make the calculations in quantum mechanics deterministic or assume continuity of space. In quantum mechanics, the mathematical framework involves the use of complex probability amplitudes, often represented by wavefunctions, to describe the state of quantum systems.
Measure theory provides a rigorous foundation for defining and analyzing probabilities, allowing for the formulation of probabilistic statements and calculations. However, the probabilistic nature of quantum mechanics arises from the fundamental principles of the theory, such as the superposition principle and the collapse of the wavefunction upon measurement.
According to the principles of quantum mechanics, the evolution of a quantum system is described by the Schrödinger equation, which is a deterministic equation. It governs the time evolution of the wavefunction and determines the probabilities of different outcomes. However, when a measurement is made, the wavefunction "collapses" into one of the possible measurement outcomes, and the outcome is probabilistic.
Regarding the continuity of space, quantum mechanics does not assume continuity at the most fundamental level. In fact, quantum mechanics allows for the possibility of discrete energy levels and quantized properties. The concept of "quantization" in quantum mechanics implies that certain physical quantities, such as energy or angular momentum, can only take on specific, discrete values rather than being continuous.
In summary, while measure theory provides a mathematical framework for calculating probabilities in quantum mechanics, it does not make the theory deterministic or assume continuity of space. The probabilistic nature and the discrete behavior of certain quantities in quantum mechanics are fundamental aspects of the theory itself.