Delve into the intricacies of Bourgain's proof of the circular maximal function theorem in this fourth installment of a lecture series presented by Jonathan Hickman from the University of Edinburgh. Explore advanced mathematical concepts and techniques as Hickman continues to unravel the complexities of this significant theorem in harmonic analysis. Gain insights into the innovative approaches and rigorous reasoning employed in Bourgain's groundbreaking work, enhancing your understanding of this fundamental result in mathematical analysis.