The conditional expectation of X given a ฯ-algebra is your best guess of X when all you know is which block of information the outcome fell in: X averaged over that block. It is itself random, because it changes with the block, but it uses only what the observer sees. The more information you have, the smaller the blocks and the closer the guess gets to X. A second page changes the probability measure by reweighting, the idea behind Girsanov's theorem and risk-neutral pricing.
Given a ฯ-algebra ๐ข, E[X | ๐ข] is the ๐ข-measurable random variable that agrees with X on average over every event in ๐ข. When ๐ข is generated by blocks this is just the block average of X, assigned to every outcome in the block. It is the best prediction of X in the least-squares sense using only the information in ๐ข. Conditioning on nothing gives the ordinary expectation, and conditioning on everything gives X itself.
Averaging the guesses gives back the plain expectation: E[E[X | ๐ข]] = E[X]. More generally, if you condition first on a fine ฯ-algebra and then on a coarser one, the result is the same as conditioning on the coarser one directly. Information that is later forgotten leaves no trace. This is why a martingale, a process whose best guess of the future is today's value, stays consistent over time.
Choose what you know about two dice and watch the guess of a value such as the larger die: it is constant on every block. Compare the squared error of the guess with the variance, which is the error of guessing without any information, and see the error shrink as the blocks get smaller. On the second page reweight samples by dQ/dP and see a standard normal turn into a shifted one without moving a point.
No. The outcomes and their values stay exactly the same; only the weights change. A new measure Q is built by giving each outcome a weight dQ/dP, and expectations under Q are P-expectations of the weighted variable. Girsanov's theorem gives that weight for whole Brownian paths. In pricing, the real-world P and the risk-neutral Q share every path and differ only in how much each path counts.