| Summary: | In 1986, J. Bourgain showed that, for a given dimension d $ ge$ 2, there exists $ rho sb{d}$ $<$ d such that any harmonic measure in $ Re sp{d}$ is supported by a set of Hausdorff dimension at most $ rho sb{d}$. === This thesis presents a detailed and comprehensive exposition of Bourgain's theorem. Formal definitions of harmonic measures and Hausdorff dimensions are provided and all the required preliminary results about these concepts are rigorously proved. Generally speaking, the proof of the theorem itself is similar to the one originally presented by Bourgain. However, a more structured approach and an increased level of details make the argument easier to follow and understand.
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