Micro-canonical ensemble and classical reality I seem to find a contradiction in the notion of probability density used by Landau and the notion of micro-canonical ensemble.
To see this, take an isolated classical system and we know experimentally that its energy lies between $E-\Delta$ and $E+\Delta$. So, we take a hypershell corresponding to these energies in phase space and say that at equilibrium, the probability density is constant in the whole shell. Now, we know that the system would be, in reality, at a fixed energy E' and the hypersurface corresponding to this energy would lie in the previous hypershell. Also, as the system is isolated, the representative point of the system would move only on this hypersurface. Now, take a point in the shell which lies outside the surface. Choose a small enough neighborhood of it that doesn't intersect the surface. Because, the probability distribution is constant, the probability of finding the system in this neighborhood is some non-zero positive number. But, as the system always remains on the surface, it never visits that neighborhood and hence the probability of finding it in that neighborhood is zero.
Am I doing something wrong?
 A: No, you're not doing anything wrong, this is all correct. As an analogy, imagine I roll a die and hide it under a cup. Since you don't know which side of the die is facing upward, you represent it with a probability distribution, with an equal probability assigned to each of the six spaces. This probability distribution doesn't change over time, in this case for the trivial reason that the die isn't moving.
You know that in reality, the die is sitting there with one particular side facing upward, and that it never "visits" any of the other sides. But unless I lift up the cup, you have no choice but to keep on thinking of it as being in a probability distribution, because you don't know which state is the true one.
With the microcanonical distribution it's the same. There is indeed one "true" energy $E'$ that doesn't change, and the system cannot visit states with any other value of $E$. But the assumption is that you don't have any way to measure the energy beyond a certain level of accuracy. So, in the analogy, the die remains hidden under the cup and you have to keep representing it with a probability distribution. 
Although many text books fail to make this clear (because it was widely misunderstood for much of the 20th century), the probability distribution doesn't represent the set of states the system can visit, it just represents experimental uncertainty about which state the system is in. It is this uncertainty that remains invariant in equilibrium.
A: In classical framework one defines an isolated system as that which is not interacting with any other system and thus whose energy is fixed.
Now the (naive) hypothesis would be that upon observation an isolated system will be found on its constant energy surface and its probability to be found in any of the states on its constant energy surface will be equal.
However as such this hypothesis is self contradictory because the very act of observation will make the system non-isolated and so in particular its energy may change by act of observation.
So one refines the hypothesis as:
Upon observation an (initially) isolated system (of energy $E$) will be found with equal probability in any of the states between energy $E-\Delta$ and $E+\Delta$ 
(Here $\Delta$ is a small number that takes into account the disturbance that your act of observation may produce.) 
