Can we always find a measure for a measurable space? I'm new to the Measure Theory. I was wondering can we always find a measure for a measurable space? It would be better to explain in details. 
 A: There's always the measure that maps every subset to $0$.
If you mean a probability measure then it also exists, pick $x\in X$ and define $f(A)=1$ if $x\in A$ and $0$ otherwise. This is called a Dirac measure.
A: A measurable space is just a pair $(X,\mathcal{M})$, where $X$ is a set and $\mathcal{M}\subseteq\mathscr{P}(X)$ is a $\sigma$-algebra on $X$. The purpose of this definition is just to identify a $\sigma$-algebra on $X$, which is the collection of all measurable subsets of $X$.
For any given set $X$, there are several ways in which we can define a measure $\mu:\mathscr{P}(X)\to[0,\infty]$. (See examples below.) For any such measure, the restriction $\mu|_{\mathcal{M}}$ will turn the measurable space $(X,\mathcal{M})$ into the measure space $(X,\mathcal{M},\mu|_{\mathcal{M}})$.
In fact, it's worth proving the following easy exercise.

If $(X,\mathcal{M},\mu)$ is a measure space and $\mathcal{N}\subseteq \mathcal{M}$ is a $\sigma$-algebra, then $\mu|_{\mathcal{N}}$ is a measure on the measurable space $(X,\mathcal{N})$.

Some important examples of measures $\mu:\mathscr{P}(X)\to[0,\infty]$:


*

*The zero measure: $\mu(A)=0$ for every $A\subseteq X$.

*The infinite measure: $\mu(A)=\infty$ for $A\ne\emptyset$ and $\mu(\emptyset)=0$.

*The counting measure:
$$
\mu(A) = \begin{cases}
|A| & \text{if $A$ is finite}, \\
\infty & \text{otherwise}.
\end{cases}
$$

*Point mass measure (or Dirac measure) of a fixed point $x\in X$:
$$\mu(A) = \begin{cases}
1 & \text{if $x\in A$}, \\
0 & \text{otherwise}.
\end{cases}$$

*$\mu(A)=\infty$ if $A$ is infinite, and $\mu(A)=0$ otherwise.

A: To construct a probability measure on any measurable space $(\Omega,\tau)$, consider the Dirac measure P:
For any $A\in \tau$
$$P(A) = \begin{cases}
1 & \text{$A\ni x$}, \\
0 & \text{otherwise}.
\end{cases}$$
Choice $x\in \Omega$ such that: $\exists B\in \tau, B\ni x $. Then $(\Omega,\tau,P)$ is a probability space, P is a probability measure.
