Probability density function

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $X_1, \ldots, X_N : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$
Let $M = (\mathbb{R}^N, \mathcal{B}(\mathbb{R}^N))$ be a D2763: Euclidean real Borel measurable space such that
 (i) $\upsilon$ is a D1743: Lebesgue measure on $\mathbb{R}^N$
A D4364: Real function $f : \mathbb{R}^N \to \mathbb{R}$ is a probability density function for $X : = (X_1, \ldots, X_N)$ if and only if \begin{equation} \forall \, E \in \mathcal{B}(\mathbb{R}^N) : \mathbb{P}(X \in E) = \int_E f(x) \, \upsilon(d x) \end{equation}
[Comment 0] : Let $\pi_1, \ldots, \pi_N$ each be a D327: Canonical set projection on $\mathbb{R}^N$ such that \begin{equation} \pi_n : \mathbb{R}^N \to \mathbb{R}, \quad \pi_n(x_1, \ldots, x_n \ldots, x_N) = x_n \end{equation} for each $n \in 1, \ldots, N$. If $E \in \mathcal{B}(\mathbb{R}^N)$, then result R4602: Element in finite cartesian product iff components in images of canonical projections shows that $X \in E$ if and only if \begin{equation} X_1 \in \pi_1 E, \quad \ldots, \quad X_N \in \pi_N E \end{equation} Thus we can write the condition also as \begin{equation} \forall \, E \in \mathcal{B}(\mathbb{R}^N) : \mathbb{P}(X_1 \in \pi_1 E, \ldots, X_N \in \pi_N E) = \int_E f(x) \, \upsilon(d x) \end{equation}
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