(Epistemic status: reflects my best understanding, but there definitely exist people who are smarter than me and disagree with me.)

There are many reasons to be radically uncertain about whether you are really you. I hope to dispel one of them today so you can sleep more soundly at night.

The second law of thermodynamics says that the universe will tend toward a state of maximum entropy, so our probability distribution over states of the world will flatten out until every state is as probable as every other state. Since the vast majority of possible states look like diffuse gasses with no interesting structure, that's what we expect the universe will become. Sure, there are some states where everything "spontaneously" turns into balloons but there are so few compared to the vast number where everything is a diffuse mist that we can essentially ignore them.

Unless the universe continues to exist for an infinite amount of time, in which case every physical state with finite probability will be visited an infinite number of times, completely at random. One state, for instance, is a Boltzmann Brain- a brain with memories and perceptions exactly identical to yours that pops out of the void for a couple of seconds, and then vanishes, never to experience again. If this happens an infinite number of times, then you are probabilistically guaranteed to be one of these random fluctuations. In that case, you should always expect to stop existing in the next moment.

This state of affairs really throws a wrench in the whole scientific programme of predicting the future. Fortunately, the idea of random fluctuations is misguided in this instance.

Consider a ball bouncing around inside of a box. The ball starts off at some position and with some momentum, each of which you know within some finite precision. If you then ask what will happen to the ball, you know that it's going to ricochet back and forth off the walls. However, the further you look forward in time, the greater uncertainty you have about where it will be and what momentum it will have. No matter how certain your measurement was at the beginning, there will be some number of bounces after which you essentially have no idea which way the ball is going. 

If you let this ball bounce around for a while and then open the box, you could think of yourself as randomly sampling a particular microstate of this ball/box system. But the ball didn't "randomly fluctuate" into that state. It followed a well-defined path. Upon observing the ball, you can make a very good guess of the path it must have taken while the box was closed in order to get from where it started to where it ended.

Here's the key: the fact that you can use a model that guesses the microstate of the system at random doesn't mean that the microstate itself starts evolving at random. The laws of physics still apply. The ball is still bouncing around. 

In the same way, the fact that we observe our actual state as being brains is evidence that a couple of seconds ago the universe was in the kind of state that would naturally lead to a brain in a few seconds. There is no rule that says "if you wait 10^10^10^10 years actually particles don't interact with each other any more and at every moment the universe is in some completely random new state." Brains made in the far future shouldn't spontaneously disappear for the same reason that brains now shouldn't disappear- because they are composed of atoms that interact in systematic, nonrandom ways.

PS: If you are familiar with inflation, another reason that we might not predict Boltzmann Brains not to exist is if random inflation bubbles are more likely to occur than random brain formation. The total mass energy of an adult human brain (around 1.4 kg) is ~10^27 GeV. The energy scale of inflation is 10^-3 times the Planck energy, or ~10^16 GeV. So assuming a Boltzmann distribution over states there will be an insanely large number of random nucleation events before any Boltzmann Brain appears. Therefore most brains will exist in the early stages of the universe before it reaches equilibrium. You may object to this model- my guess is that any objection is comparable to my objection above. The specific mechanisms by which brains form or inflation occurs are more precise than just "probability of state given by Boltzmann factor."