Let $V \colon [0,1]\times \mathbb R^d \to \mathbb R^d$ be a Borel vector field which is globally bounded, $V \in L^\infty$. 


I am looking for a reference for the following result (which I suppose it is true and already been proved somewhere). 



> 
> The set of integral curves of $V$, i.e. 
> 
> 
> $\displaystyle \mathcal C :=\big\{ \gamma \in C([0,1];\mathbb R^d): \gamma(s)-\gamma(t) - \int\_t^s V(\tau,\gamma(\tau)) d\tau = 0, \quad \forall 0\le t \le s \le 1\big\}$
> 
> 
> is a Borel set in the space of continuous path $C([0,1];\mathbb R^d)$ endowed with the topology inherited from the $\sup$ norm. 
> 
> 
> 


*[The integral curves are defined for all times being the vector field bounded]*


It is immediate to show that for *continuous* vector fields the statement holds (the set $\mathcal C$ is closed) but I do not find anything for the general case.



Set of integral curves of a vector field
Actually this follows plainly as a consequence of an important classical result in descriptive set theory, namely, the set of all Borel real-valued functions on a metric space is the smallest class containing the continuous functions, and closed under point-wise convergence, that is, the Baire class. (And, of course, nothing changes, if you consider $\mathbb{R}^d$-valued maps instead, or also, maps valued in the unit ball of $\mathbb{R}^d$).


A suitable reference is e.g. A. S. Kechris' *Classical Descriptive Set Theory*, ch. 11, or [this paper](http://www.ams.org/journals/tran/1974-194-00/S0002-9947-1974-0362270-7/S0002-9947-1974-0362270-7.pdf) by R. W. Hansell available on-line, that contains all ingredients. (Alternatively, you may replace "Borel vector field" with "Baire vector field" in the statement).


For any bounded Borel map $V:[0,1]\times\mathbb{R}^d\to\mathbb{R}^d$ we can consider the operator $F\_V$ on the Banach space $E:=C^0([0,1],\mathbb{R}^d), \|\cdot\|\_\infty$ defined by $F\_V( \gamma)(t)=\int\_0^tV(\tau,\gamma(\tau))d\tau$ for any $\gamma\in E$ and $t\in[0,1]$. Two simple facts easily follow by bounded convergence:


* If $V$ is continuous on $[0,1]\times\mathbb{R}^d$, then $F\_V$ is continuous on $E$. Indeed, if $\gamma\_n\to\gamma$ in $E$
$$\big\|F\_V(\gamma )-F\_V(\gamma\_n)\big\|\_\infty\le\int\_0^1\big\|V(\tau,\gamma(\tau))-V(\tau,\gamma\_n(\tau))\big\| d\tau=o(1).$$
* If $V\_n$ converges point-wise to $V$ on $[0,1]\times\mathbb{R}^d$, then $F\_{V\_n}$ converges point-wise to $F\_V$ on $E$. Indeed, if $V\_n\to V$ point-wise, then for any $\gamma\in E$
$$\big\|F\_V(\gamma )-F\_{V\_n}(\gamma)\big\|\_\infty\le\int\_0^1\big\|V(\tau,\gamma(\tau))-V\_n(\tau,\gamma(\tau))\big\| d\tau=o(1).$$


As a consequence, the set of all bounded Borel maps $V:[0,1]\times\mathbb{R}^d\to\mathbb{R}^d$ such that $F\_V$ is a Borel map on $E$ contains all bounded continuous functions and it is closed under point-wise convergence. Therefore it is the whole class of bounded Borel maps.


To conclude, let's denote ${\mathbb 1}$ and $\mathrm{ev\_0}$ the identity map on $E$, respectively, the evaluation at $0$ (seen as a bounded, linear rank-one operator on $E$). The set
$$\mathcal{C}=\big\{\gamma\in E : \gamma(t)-F\_V(\gamma)(t)-\gamma(0)=0,\; \forall\, t\in[0,1] \big\},$$
which is the zero-set of the Borel map ${\mathbb 1}-F\_V-\mathrm{ev\_0}$, is therefore a Borel set. 


**Rmk**: From the above argument also follows by induction that, more precisely, if $V$ is in the Baire class $\alpha$ for a countable ordinal $\alpha$, then so is $F\_A$ on $E$, so that $\mathcal{C}$ is in the corresponding Borel set hierarchy.


