Take an arbitrary infinite binary sequence $\omega$. The interesting case is when $\omega$ is not computable. Is there a computable (semi-)measure $\mu$ such that sequence $\omega$ is $\mu$-random in the sense of Martin-Lf?Or $\mu$-random is some weaker sense?Clearly, $\omega$ is $\mu$-random for $\mu$-almost all sequences. So the question is what happens with the sequences from that set of measure zero. 
Is an infinite incomputable sequence random wrt a computable measure?
computability;randomness
Mucknik, Semenov and Uspensky showed that there are sequences which are not Martin-Lf random for any computable measure.  They call all other sequences (which are Martin-Lf random for some computable measure) natural sequences.Andrei A. Muchnik, Alexei Semenov, and Vladimir Uspensky. Mathematical metaphysics of randomness. Theoretical Computer Science, 207(2):263317, 1998.This topic is also discussed in Laurent Bienvenu and Christopher Porter. Effective randomness, strong reductions and Demuths theorem.Jan Reimann and Theodore A. Slaman. Probability measures and effective randomness.Jan Reimann and Theodore A. Slaman. Measures and their random reals.Here is a quick proof (which may be different from the above references):  Let $\{U_i\}$ be a (non-effective) enumeration of all $\Sigma^0_1$ sets making up some level of the universal Martin-Lf test for some computable measure $\mu$.  Each $U_i$ is dense and open, so by the Baire category theorem (countable intersection of dense open sets is dense), the intersection $\bigcap_i U_i$ is nonempty.  Any sequence in this intersection cannot be Martin-Lf random for any computable measure $\mu$.The same can be done for Schnorr randomness (which is weaker than Martin-Lf randomness). Let $\{U_i\}$ be the collection of all dense $\Sigma^0_1$ sets of computable $\mu$-measure for some computable measure $\mu$.UPDATE Oct 23, 2013:I realize there are three things to add (which I am sure are all well-known and they are probably in the above mentioned papers).My above proof actually shows that 1-generics cannot be Martin-Lf random for a computable measure, since 1-generics are in all dense $\Sigma^0_1$ sets.The K-trivials are not Martin-Lf random on any computable measure.  A sequence $X \in \{0,1\}^\mathbb{N}$ is low-for-random (a.k.a. $K$-trivial) if $\text{MLR}^X = \text{MLR}$. It can be shown that this property also extends to all computable $\mu$.  That is, if $X$ is K-trivial, then $\text{MLR}_{\mu}^X = \text{MLR}_{\mu}$ for all computable $\mu$. Hence, the non-computable $K$-trivials cannot be random for any computable measure.  For if a $K$-trivial $X$ is random on a computable measure $\mu$ then $X$ is random relative to $X$.  This means $X$ is an atom.  However, only computable points can be atoms of computable measures.The set of all natural sequences is precisely the set of sequences truth-table reducible to a Martin-Lf random (on the fair-coin measure).  (The main idea is that every computable measure is the push-forward on an a.e. computable map, and the Martin-Lf randoms on the measure $\mu$ are exactly the sequences pushed over to the new measure.  The reason this computation is truth-table involves using layerwise computability.)