Is Murphy's law true? I was reading this question which got me thinking about the old saying "whatever can go wrong will eventually go wrong", aka Murphy's law and I began to think of ways to prove/disprove it.
I found this proof of 'Murphy's law' but here they seem to assume that the probability of the event remains non-zero (the assumption is $\mathbb P (A \mid A_n^c)>\epsilon$). However, as far as I know, a 'possible event' does not need to have a non-zero probability, it only needs to be a measurable event, and can have measure zero for that matter. 
For example if we have a probability space $(\mathbb R, \mathcal B, \mathbb P)$ where $\mathbb P$ is the Lebesgue measure, then any singleton $\lbrace a \rbrace$ has measure zero. If we keep drawing singletons, will we eventually draw some particular singleton? I tried to think in terms of Borel-Cantelli Lemma and define an event something like $E_n =  a \in\lbrace d_1, d_2, \dots,d_n \rbrace $ where $d_i$ is draw $i$ but is seems that for any $n$, $\mathbb P(E_n)=0$ so it seems to contradict 'Murphy's law'. 
So I guess my main question is: are there any general statements to be made about infinite sequences of events where the events have probability zero? Does it event exist an experiment where $\mathbb P (E_n)=0$ but $\mathbb P(E_n \text{ at least once as } n\rightarrow \infty) =1$?
 A: By countable additivity, if all $\mathbb P(E_n) = 0$ then $\mathbb P(\bigcup_{n=1}^\infty E_n) = 0$.
Where you can get nonzero probabilities is for the union of uncountably
many events, each of probability $0$.  
A: If you are thinking about physical (observable) reality, which is usually what Murphy's Law refers to, then we need to be very specific about the (mathematical) structure of reality. Is it discrete or continuous time? Deterministic or random? There are too many unresolved scientific and philosophical questions to declare Murphy's Law as actually true or false.
Some believe that all possible universes exist in some sense, which would mean that everything that is possible does eventually happen (somewhere). However, if we restrict ourselves to the future of our (observable) universe, then there may be possible things that do not occur. Of course it depends on the detailed (and unresolved) structure of our particular universe.
For example, I could have typed a bunch of nonsense, a completely random string of letters and submitted it as my answer here. However, instead, I posted a meaningful, reasonable and honest answer -- of course, as the reader, you may disagree with that assertion!
Given that I have only one chance to submit a first answer, we now know of something that was possible, but did not occur, and therefore Murphy's Law is false. Although, there are deep philosophical issues left unresolved here.
Of course this is not a mathematical answer and so it will probably get downvoted. Just in case, an informal mathematical example follows.
===
Take the set of functions that have $\mathbb{R}$ as their range and that $f:[0,\infty)\rightarrow\mathbb{R}$. Surely there are uncountably many, and if we are to select one with some random rule, then each outcome has probability zero. If we think of $f(t)$ as the state of a process at time $t$, then the probability of any particular individual outcome at any particular time is zero (given an appropriately chosen random rule). Note that we aren't assuming any kind of continuity or smoothness property for the functions. However, for any $x\in\mathbb{R}$, eventually there will be a $t$ such that $f(t)=x$. Of course, we need more details to determine how fast that happens, but all process realizations do hit every real number. In that sense you could say that eventually all possible outcomes do occur even though each outcome as probability zero.
===
A more formal example is Brownian Motion. Brownian motion hits any arbitrary real number $x$ in finite time with probability one. However, the expected time to hit $x$ is infinite. So as time goes on, a Brownian Motion path will eventually hit every real number.
