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  "body": "\n## Proof\n\nWe write $e^{tS}$ as a product and argue by induction on the number of factors. Truncation and monotone convergence handle infinite values.\n\n1. Let $X_1, \\dots, X_n$ be [[independent|0XCE070V8B]] real random variables on $(\\Omega, \\mathcal{F}, P)$, and let $S = X_1 + \\dots + X_n$. Fix $t \\in \\R$.\n2. For each $j$, let $W_j = e^{tX_j}$, a random variable with values in $(0, \\infty)$. By the [[exponential of a sum|0E8HNVBCSP]], $e^{tS} = W_1 W_2 \\cdots W_n$.\n3. By the definition of the [[expected value|0DCT0QZVQ1]] of a nonnegative variable, $M_{X_j}(t) = E[W_j] \\in (0, \\infty]$ and $M_S(t) = E[W_1 \\cdots W_n]$.\n4. For $2 \\le m \\le n$, let $\\mathcal{P}_m$ be the family of events $\\bigcap_{j \u003c m} \\set{X_j \\in B_j}$ with Borel sets $B_j \\subseteq \\R$. Let $\\mathcal{G}_m$ be the σ-algebra generated by $\\mathcal{P}_m$.\n5. The intersection of two such events is $\\bigcap_{j \u003c m} \\set{X_j \\in B_j \\cap B_j'}$, again in $\\mathcal{P}_m$. So $\\mathcal{P}_m$ is a [[π-system|4W9NM803BB]].\n6. Fix a Borel set $B$ and let $C = \\set{X_m \\in B}$. For $A = \\bigcap_{j \u003c m} \\set{X_j \\in B_j}$ in $\\mathcal{P}_m$, the definition of independence with $J = \\set{1, \\dots, m}$ gives $P(A \\cap C) = P(X_m \\in B) \\prod_{j \u003c m} P(X_j \\in B_j)$.\n7. The same definition with $J = \\set{1, \\dots, m - 1}$ gives $\\prod_{j \u003c m} P(X_j \\in B_j) = P(A)$. So $P(A \\cap C) = P(A) P(C)$ for each $A \\in \\mathcal{P}_m$.\n8. Let $\\mathcal{D}$ be the family of events $A$ with $P(A \\cap C) = P(A) P(C)$. It contains $\\Omega$.\n9. If $A \\subseteq A'$ are in $\\mathcal{D}$, then $P((A' \\setminus A) \\cap C) = P(A' \\cap C) - P(A \\cap C) = (P(A') - P(A)) P(C)$. This is $P(A' \\setminus A) P(C)$, so $A' \\setminus A \\in \\mathcal{D}$.\n10. If $A_1 \\subseteq A_2 \\subseteq \\cdots$ are in $\\mathcal{D}$ with union $A$, then $A_k \\cap C$ increases to $A \\cap C$. By [[continuity of probability|0X8HBKCSYB]], $P(A \\cap C) = \\lim_k P(A_k) P(C) = P(A) P(C)$.\n11. By steps 8 to 10, $\\mathcal{D}$ is a [[Dynkin system|69460Z6DMW]] that contains $\\mathcal{P}_m$. By [[Dynkin's π-λ theorem|BBHM1A36M7]], $\\mathcal{G}_m \\subseteq \\mathcal{D}$.\n12. Let $V_m = W_1 \\cdots W_{m-1}$. For $j \u003c m$, the function $x \\mapsto e^{tx}$ is continuous, so the preimage of a Borel set under it is Borel. Each event $\\set{W_j \\in D}$ with $D$ Borel has the form $\\set{X_j \\in B_j}$ and lies in $\\mathcal{G}_m$.\n13. So each $W_j$ with $j \u003c m$ is [[measurable|YYNCJSR0PJ]] with respect to $\\mathcal{G}_m$. A product of measurable functions is measurable, so $V_m$ is $\\mathcal{G}_m$-measurable.\n14. For an integer $N \\ge 1$, let $V_{m,N} = \\min(V_m, N)$ and $W_{m,N} = \\min(W_m, N)$. Each event $\\set{V_{m,N} \\in D}$ with $D$ Borel lies in $\\mathcal{G}_m$.\n15. Each event $\\set{W_{m,N} \\in D'}$ with $D'$ Borel has the form $\\set{X_m \\in B}$ for a Borel set $B$. By step 11, $V_{m,N}$ and $W_{m,N}$ are [[independent|0E9502NT92]].\n16. Both are bounded, so they are integrable. By the [[product rule for independent variables|05AW1G0WE1]], $E[V_{m,N} W_{m,N}] = E[V_{m,N}] \\, E[W_{m,N}]$.\n17. As $N \\to \\infty$, $V_{m,N}$ increases to $V_m$, $W_{m,N}$ increases to $W_m$, and $V_{m,N} W_{m,N}$ increases to $V_m W_m$. All are nonnegative.\n18. By the [[monotone convergence theorem|G2S6H789RG]], the three expectations in step 16 converge in $[0, \\infty]$ to $E[V_m W_m]$, $E[V_m]$ and $E[W_m]$.\n19. The sequences $E[V_{m,N}]$ and $E[W_{m,N}]$ are nondecreasing and positive. So their product tends to $E[V_m] \\, E[W_m]$ in $(0, \\infty]$. Hence $E[V_m W_m] = E[V_m] \\, E[W_m]$.\n20. Since $V_m W_m = V_{m+1}$ with $V_{n+1} = W_1 \\cdots W_n$, step 19 gives $E[V_{m+1}] = E[V_m] \\, E[W_m]$ for $2 \\le m \\le n$.\n21. By induction on $m$, starting from $V_2 = W_1$, we get $E[V_{m+1}] = E[W_1] \\cdots E[W_m]$ for $1 \\le m \\le n$.\n22. For $m = n$, steps 2, 3 and 21 give $M_S(t) = M_{X_1}(t) \\, M_{X_2}(t) \\cdots M_{X_n}(t)$ in $(0, \\infty]$.\n\n## Idea\n\nThe exponential turns the sum $S$ into the product of the factors $e^{tX_j}$. Independence lets the expectation of a product split into the product of the expectations. The π-λ theorem makes the first $m - 1$ factors, taken together, independent of the next one, and truncation reduces infinite values to bounded variables.\n",
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