Construct a measure space $(X, \mathcal{A}, \mu)$ with $\mu$ probability measure (that is, $\mu(X) = 1$) and a function $f: X \to \mathbb{R}$ that is measurable with respect to $\mathcal{A}$ and $\mathcal{B}(\mathbb{R})$ (the Borel $\sigma$-algebra on $\mathbb{R}$) such that there is a $\lambda > 0$ with $$\forall n \in \mathbb{Z}_+, \mu( \{x \in X \ \mid \ f(x) = n \} ) = \frac{\lambda^n \cdot e^{-\lambda}}{n!}.$$
Also find a function non-decreasing, right continuous, bounded function with $\displaystyle \lim_{x \to -\infty} (F(x)) = 0$ and $\displaystyle \lim_{x \to \infty}(F(x)) = \infty$ such that $$\mu_F = f_*\mu, $$ where $f_*\mu$ is the push-forward measure of $\mu$ by $f$ and $\mu_F$ is the measure generated by $F$, that is, $\mu_F((a,b]) = F(b) - F(a), \forall a < b, a,b \in \mathbb{R}$.
I do not know how to begin constructing such a function. I will remark that I have not taken any course in probability theory and I am taking the measure theory now, which in this particular case provides difficulty.
To construct a probability space and a random variable with Poisson distribution take $X=\mathbb N$, $\mathcal A$ to be the power set and define $P$ by $P(E)=\sum_{n \in E} e^{-\lambda}\frac {\lambda^{n}} {n!}$. Define $f(n)=n$ for all $n$. Define $F$ by $F(x)=\sum_{n \leq x} P\{n\}$. This satisfies the required properties.